#6622810

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1395 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,18,22,27)(19,23,31,39)(20,32,36,42)(21,25,33,26)(24,34,40,35)(28,43,45,47)(29,38,30,41)(37,46,48,44);(2,6)(7,8)(10,14)(11,12)(17,19)(18,23)(20,28)(21,30)(22,31)(24,37)(25,41)(26,38)(27,39)(29,33)(32,43)(34,46)(35,44)(36,45)(40,48)(42,47);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,33,34,22,36,21,35)(18,24,25,42,27,40,26,32)(19,28,29,46,31,45,30,44)(23,37,41,47,39,48,38,43);(1,4,5,3)(2,8,6,7)(11,12)(15,16);(9,12,13,11)(10,16,14,15)(17,21,22,33)(18,25,27,26)(19,29,31,30)(20,34,36,35)(23,38,39,41)(24,32,40,42)(28,44,45,46)(37,47,48,43)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6530006);