#6614953

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1469 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:

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

Polycyclic presentation:

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

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, 6522149);