#181227

order	512
nilpotency class	3
isoclinism	isoclinic class of #181131
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#95
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1308 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9)(4,12)(5,8)(6,11)(7,13)(10,15)(14,16)(17,18)(19,27,29,23)(20,24)(21,26)(22,25)(28,37,31,35)(30,41,44,36)(32,38)(33,39)(34,40)(42,46)(43,48,45,47);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(17,19,21,30,22,29,34,44)(18,23,26,36,25,27,40,41)(20,31,33,45,32,28,42,43)(24,37,39,48,38,35,46,47);(3,10)(6,14)(9,15)(11,16)(17,20)(18,24)(19,28)(21,33)(22,32)(23,35)(25,38)(26,39)(27,37)(29,31)(30,43)(34,42)(36,47)(40,46)(41,48)(44,45);(20,32)(24,38)(28,31)(33,42)(35,37)(39,46)(43,45)(47,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=hi,g^2=i,b^a=be,e^a=eh,g^a=ghi,c^b=ch,f^b=fi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317810, #72318533, #72318627
maximal normal subgroup of	#72317810, #72318533, #72318627

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88423);