#495431

order	512
nilpotency class	4
isoclinism	isoclinic class of #480482
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

972 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)(3,7)(4,5)(6,8)(9,10,16,18,15,21,12,27)(11,22,24,38,26,34,30,37)(13,19,23,32,14,20,28,36)(17,31,33,29,35,39,40,25);(2,5)(6,7)(9,11,23,39,15,26,28,31)(10,17,32,37,21,35,36,38)(12,24,14,29,16,30,13,25)(18,33,19,34,27,40,20,22);(1,3,4,8)(2,6,5,7)(9,12,15,16)(10,18,21,27)(11,24,26,30)(13,23,14,28)(17,33,35,40)(19,36,20,32)(22,37,34,38)(25,39,29,31);(9,13,15,14)(10,19,21,20)(11,25,26,29)(12,28,16,23)(17,34,35,22)(18,32,27,36)(24,31,30,39)(33,37,40,38)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,d^c=di,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402627);