#72237397

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#21746
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂⁴
prime radical	#4379699	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#534 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

3896 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=ef,e^2=i,b^a=be,c^a=cghij,c^b=cghj,d^a=dj,d^b=dgh,e^a=ej,e^b=eh,e^c=ei,f^a=fhij,f^b=fh,f^d=fi,f^e=fj,g^b=gij,h^b=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237401, #72238432, #72238440
maximal normal subgroup of	#72237401, #72238432, #72238440

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