#393235

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300431);