#181180

order	512
nilpotency class	3
isoclinism	isoclinic class of #181129
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

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

1686 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88376);