#6507312

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	#25125	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1654 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)(3,6)(4,5)(7,8)(17,18,23,27)(19,28,31,39)(20,25,32,41)(21,26,22,37)(24,35,40,30)(29,45,33,48)(34,43,36,44)(38,46,42,47);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(20,32)(25,41)(29,33)(34,36)(38,42)(43,44)(45,48)(46,47);(9,11)(10,13)(12,16)(14,15)(17,19,21,35,23,31,22,30)(18,24,37,39,27,40,26,28)(20,33,36,47,32,29,34,46)(25,42,43,45,41,38,44,48);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20,23,32)(18,25,27,41)(19,29,31,33)(21,36,22,34)(24,38,40,42)(26,44,37,43)(28,45,39,48)(30,47,35,46);(10,14)(13,15)(17,21,23,22)(18,26,27,37)(19,30,31,35)(20,34,32,36)(24,39,40,28)(25,43,41,44)(29,46,33,47)(38,45,42,48)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei,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, 6414508);