#358227

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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 ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1670 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,g^b=gi,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, 265423);