#6628080

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,8)(4,6)(5,7)(9,10,13,14)(11,15,12,16)(17,18,21,25)(19,27,29,23)(20,31,32,40)(22,35,34,26)(24,36,39,33)(28,43,44,48)(30,38,46,42)(37,45,41,47);(1,3,5,4)(2,6,7,8)(10,14)(11,12)(17,19,21,29)(18,23,25,27)(20,28,32,44)(22,30,34,46)(24,37,39,41)(26,38,35,42)(31,48,40,43)(33,45,36,47);(9,11,13,12)(10,15,14,16)(17,20,21,32)(18,24,25,39)(19,28,29,44)(22,36,34,33)(23,37,27,41)(26,40,35,31)(30,47,46,45)(38,43,42,48);(1,4,5,3)(2,6,7,8)(11,12)(15,16)(18,25)(23,27)(24,39)(26,35)(31,40)(37,41)(38,42)(43,48);(9,12,13,11)(10,16,14,15)(19,29)(23,27)(28,44)(30,46)(37,41)(38,42)(43,48)(45,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,c^2=hi,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6535276);