#494446

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
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	C₂⁵	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1486 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)(9,10,14,19,30,17,28,37,15,21,13,18,12,20,29,36)(11,22,25,33,23,39,31,16,27,34,24,40,26,38,32,35);(1,2,3,4)(9,11)(10,16)(12,23)(13,31)(14,32)(15,27)(17,33)(18,38)(19,39)(20,40)(21,35)(22,37)(24,28)(25,29)(26,30)(34,36);(6,8);(9,12,15,30)(10,17,21,20)(11,23,27,26)(13,28,14,29)(16,33,35,40)(18,36,19,37)(22,38,34,39)(24,32,25,31)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,d^2=i,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^b=eh,g^b=gh,h^b=hi

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, 401642);