#495401

order	512
nilpotency class	4
isoclinism	isoclinic class of #480460
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

990 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,d^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,g^b=gh,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, 402597);