#376126

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
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	#23386	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1610 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)(9,10,14,19,15,20,13,18)(11,21,24,31,25,30,23,16)(12,17,28,32,26,33,27,34)(22,35,38,39,36,40,37,29);(5,6)(7,8)(9,11,13,23,15,25,14,24)(10,16,19,21,20,31,18,30)(12,22,27,37,26,36,28,38)(17,29,32,35,33,39,34,40);(2,4)(6,8)(9,12)(10,17)(11,22)(13,27)(14,28)(15,26)(16,29)(18,34)(19,32)(20,33)(21,35)(23,37)(24,38)(25,36)(30,40)(31,39);(10,18,20,19)(12,26)(16,30,31,21)(17,32,33,34)(22,36)(27,28)(29,35,39,40)(37,38)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=eh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283322);