#393204

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

1152 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)(11,21)(12,16)(13,26)(14,29)(15,20)(17,23)(18,28)(19,30)(22,33)(24,37)(25,32)(27,34)(31,38)(35,36)(39,40);(5,7,8,6)(9,11,14,12,15,23,30,25)(10,16,19,21,20,32,29,17)(13,22,27,24,28,36,40,38)(18,31,34,35,26,37,39,33);(1,2,3,4)(9,12,14,23,15,25,30,11)(10,17,19,16,20,21,29,32)(13,24,27,36,28,38,40,22)(18,33,34,31,26,35,39,37);(9,13,14,27,15,28,30,40)(10,18,19,34,20,26,29,39)(11,22,12,24,23,36,25,38)(16,31,21,35,32,37,17,33)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=f,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,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, 300400);