#494448

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,e^b=eh,g^b=gh,h^b=hi,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, 401644);