#495429

order	512
nilpotency class	4
isoclinism	isoclinic class of #480476
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×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

980 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)(9,10,15,18,14,17,12,25)(11,20,22,38,24,36,29,37)(13,19,26,35,28,34,21,31)(16,30,33,27,32,39,40,23);(2,5)(6,7)(9,11,21,39,14,24,26,30)(10,16,31,37,17,32,35,38)(12,22,13,27,15,29,28,23)(18,33,34,36,25,40,19,20);(1,3,4,8)(2,6,5,7)(10,17)(16,32)(18,25)(19,34)(20,36)(31,35)(33,40)(37,38);(9,12,14,15)(10,18,17,25)(11,22,24,29)(13,26,28,21)(16,33,32,40)(19,31,34,35)(20,37,36,38)(23,39,27,30)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=fi,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,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, 402625);