#6508351

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6415547);