#357737

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2730 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)(9,10,14,18,15,19,13,27)(11,16,21,32,22,33,28,31)(12,23,26,17,25,36,29,35)(20,34,38,39,37,40,24,30);(1,2,3,4)(9,11)(10,16)(12,24)(13,28)(14,21)(15,22)(17,34)(18,32)(19,33)(20,26)(23,30)(25,38)(27,31)(29,37)(35,40)(36,39);(6,8)(9,12)(10,17)(11,20)(13,29)(14,26)(15,25)(16,30)(18,36)(19,35)(21,38)(22,37)(23,27)(24,28)(31,40)(32,34)(33,39);(12,25)(17,35)(20,37)(23,36)(24,38)(26,29)(30,39)(34,40)

Polycyclic presentation:

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

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, 264933);