#512026

order	512
nilpotency class	4
isoclinism	isoclinic class of #511802
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

875 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,h^a=hi,c^b=ci,f^c=fi,g^c=ghi,f^d=fh,g^d=gi,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419222);