#6611512

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1526 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,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, 6518708);