#393173

order	512
nilpotency class	3
isoclinism	isoclinic class of #32944
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fi,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,c^b=ci,g^d=gi

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, 300369);