#357806

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

2718 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)(9,10,14,18,15,19,13,28)(11,20,24,31,22,37,23,16)(12,17,26,34,27,35,29,33)(21,36,25,40,38,32,39,30);(1,2,3,4)(9,11,13,23,15,22,14,24)(10,16,18,20,19,31,28,37)(12,25,26,38,27,39,29,21)(17,32,33,40,35,36,34,30);(6,8)(9,12)(10,17)(11,21)(13,29)(14,26)(15,27)(16,30)(18,34)(19,35)(20,36)(22,38)(23,39)(24,25)(28,33)(31,40)(32,37);(11,22)(16,31)(20,37)(21,38)(23,24)(25,39)(30,40)(32,36)

Polycyclic presentation:

2^9|a^2=h,b^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,d^b=di,e^c=ei,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, 265002);