#181295

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

1246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=hi,f^2=i,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88491);