#6624813

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1379 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)(9,10)(11,15)(12,16)(13,14)(17,18,23,28)(19,29,31,40)(20,34,37,41)(21,26,22,27)(24,33,39,32)(25,36,42,35)(30,38,44,48)(43,46,47,45);(2,6)(7,8)(10,14)(11,12)(17,19,22,33,23,31,21,32)(18,24,26,40,28,39,27,29)(20,30,35,45,37,44,36,46)(25,38,34,47,42,48,41,43);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,21,36,23,37,22,35)(18,25,27,41,28,42,26,34)(19,30,32,46,31,44,33,45)(24,38,29,43,39,48,40,47);(1,4,5,3)(2,8,6,7)(11,12)(15,16)(19,31)(24,39)(29,40)(30,44)(32,33)(38,48)(43,47)(45,46);(9,12,13,11)(10,16,14,15)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,35,37,36)(24,40,39,29)(25,41,42,34)(30,45,44,46)(38,43,48,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=fgh,e^2=hi,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^b=eh,d^c=dh,e^c=ei,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, 6532009);