#6508608

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1461 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,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, 6415804);