#72233799

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#13967
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	#6491103	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

4094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=j,d^2=i,f^2=i,b^a=bh,c^a=cf,c^b=ce,d^b=dhij,d^c=di,e^a=ei,e^d=ej,f^b=fij,f^c=fi,g^c=gi,h^c=hj

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233808, #72234085, #72234099
maximal normal subgroup of	#72233808, #72234085, #72234099

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.7);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);