#393237

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

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,c^b=ci

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, 300433);