#6556805

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1331 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)(3,7)(4,8)(5,6)(17,18,23,27)(19,28,30,44)(20,33,37,48)(21,26,22,38)(24,31,40,32)(25,35,42,36)(29,41,34,39)(43,45,47,46);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,32,23,30,21,31)(18,24,26,28,27,40,38,44)(20,34,36,45,37,29,35,46)(25,41,33,47,42,39,48,43);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,21,35,23,37,22,36)(18,25,38,48,27,42,26,33)(19,29,31,45,30,34,32,46)(24,39,44,47,40,41,28,43);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(19,30)(24,40)(28,44)(29,34)(31,32)(39,41)(43,47)(45,46);(1,4,5,3)(2,8,6,7)(11,15)(14,16)(17,21,23,22)(18,26,27,38)(19,31,30,32)(20,35,37,36)(24,28,40,44)(25,33,42,48)(29,45,34,46)(39,43,41,47)

Polycyclic presentation:

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

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, 6464001);