#72238882

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#386
commutator	C₂²×C₄	C₂²×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂⁴
socle	C₂²	C₂×#1437
prime radical	C₂²×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

5185 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,9,11,6)(4,7,13,16)(5,8)(10,15)(12,14)(17,18,22,26,23,19,21,25)(20,28,30,27,31,32,29,24);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,29)(30,32);(1,4)(2,7,8,16)(3,10,11,12)(5,13)(6,14)(9,15)(17,20,21,29,23,31,22,30)(18,24,26,28,19,27,25,32);(2,8)(4,13)(6,9)(10,12)(17,21,23,22)(18,25,19,26)(20,29,31,30)(24,32,27,28)

Polycyclic presentation:

2^10|a^2=e,b^2=i,c^2=ef,d^2=j,e^2=j,g^2=j,b^a=bg,c^a=cgh,c^b=ch,d^a=di,d^c=di,e^b=ei,e^c=ei,f^a=fij,f^b=fj,f^c=fi,g^a=gij,g^b=gj,g^c=gj,h^c=hij

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238883, #72238951
maximal normal subgroup of	#72238883, #72238951
maximal quotient of	#72238902, #72238982

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 3, F.5*F.6);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);