#508186

order	512
nilpotency class	4
isoclinism	isoclinic class of #508078
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

922 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)(3,6)(4,5)(7,8)(9,10)(11,19)(12,23)(13,27)(14,16)(15,21)(17,26)(18,28)(20,37)(22,33)(24,32)(25,34)(29,40)(30,36)(31,39)(35,38);(2,5)(6,8)(11,20)(13,28)(15,31)(18,27)(19,37)(21,39)(22,29)(24,38)(25,30)(32,35)(33,40)(34,36);(3,7)(6,8)(9,11)(10,15)(12,24)(13,29)(14,22)(16,32)(17,33)(18,35)(19,23)(20,28)(21,26)(25,39)(27,31)(30,38)(34,37)(36,40);(1,3,4,7)(2,6,5,8)(10,16)(15,32)(17,23)(18,27)(19,33)(31,35)(34,36)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415382);