#6611506

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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

1524 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)(24,31)(29,42)(30,38)(32,43)(33,39)(34,40)(35,41)(37,44)(45,47)(46,48);(2,5)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,31,23,32,21,30)(18,24,26,28,27,38,36,43)(20,29,33,44,34,45,35,46)(25,37,41,42,40,48,39,47);(1,3,4,8)(2,6,5,7)(11,16)(13,14)(17,20)(18,25)(19,29)(21,35)(22,33)(23,34)(24,37)(26,41)(27,40)(28,42)(30,46)(31,44)(32,45)(36,39)(38,48)(43,47);(9,11)(10,14)(12,16)(13,15)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,34,35)(24,28,38,43)(25,39,40,41)(29,44,45,46)(37,47,48,42);(11,16)(13,14)(20,34)(25,40)(29,45)(33,35)(37,48)(39,41)(42,47)(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,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, 6518702);