#6595497

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1284 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)(3,7,4,8)(9,10,12,14)(11,13,15,16)(17,18,22,26)(19,28,30,23)(20,33,34,42)(21,25,37,27)(24,35,40,36)(29,44,46,48)(31,41,32,39)(38,45,43,47);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,30)(18,23,26,28)(20,29,34,46)(21,32,37,31)(24,38,40,43)(25,41,27,39)(33,48,42,44)(35,47,36,45);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,21,36,22,34,37,35)(18,24,27,33,26,40,25,42)(19,29,32,45,30,46,31,47)(23,38,39,48,28,43,41,44);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,15)(14,16)(19,30)(20,34)(23,28)(24,40)(31,32)(33,42)(35,36)(39,41);(1,4,5,3)(2,8,6,7)(11,15)(13,16)(17,21,22,37)(18,25,26,27)(19,31,30,32)(20,35,34,36)(23,39,28,41)(24,33,40,42)(29,45,46,47)(38,44,43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6502693);