#392916

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1462 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,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, 300112);