#6622923

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	C₂×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1484 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)(4,6)(5,8)(10,14)(15,16)(17,18,23,27)(19,24,29,38)(20,32,33,46)(21,26,22,36)(25,34,40,35)(28,41,44,48)(30,39,31,45)(37,42,47,43);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,19)(18,24)(20,28)(21,30)(22,31)(23,29)(25,37)(26,39)(27,38)(32,41)(33,44)(34,42)(35,43)(36,45)(40,47)(46,48);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,20,22,35,23,33,21,34)(18,25,26,32,27,40,36,46)(19,28,31,43,29,44,30,42)(24,37,39,41,38,47,45,48);(1,4,5,3)(2,6,8,7)(11,12)(15,16)(19,29)(20,33)(24,38)(25,40)(30,31)(32,46)(34,35)(39,45);(9,12,13,11)(10,16,14,15)(17,21,23,22)(18,26,27,36)(19,30,29,31)(20,34,33,35)(24,39,38,45)(25,32,40,46)(28,42,44,43)(37,41,47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fg,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,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, 6530119);