#369160

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1902 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)(5,6,8,7)(9,10)(11,22)(12,17)(13,30)(14,20)(15,21)(16,24)(18,29)(19,28)(23,35)(25,34)(26,37)(27,40)(31,36)(32,38)(33,39);(5,7,8,6)(9,11,14,24,15,26,28,25)(10,16,19,22,21,34,20,37)(12,27,13,23,29,38,31,39)(17,35,36,40,18,33,30,32);(1,2,3,4)(9,12,28,31,15,29,14,13)(10,17,19,36,21,18,20,30)(11,23,24,38,26,39,25,27)(16,32,37,33,34,40,22,35);(9,13,14,29,15,31,28,12)(10,18,19,30,21,17,20,36)(11,23,24,38,26,39,25,27)(16,33,22,32,34,35,37,40)

Polycyclic presentation:

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

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, 276356);