#6605658

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1760 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,23,28,21,26)(19,29,31,38,32,42,36,24)(20,25,33,39,34,40,35,41)(30,43,46,48,45,47,44,37);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,37)(26,29)(27,42)(28,38)(33,44)(34,45)(35,46)(39,47)(40,48)(41,43);(2,6)(5,8)(10,14)(11,15)(17,20,23,34)(18,25,28,40)(19,30,32,45)(21,35,22,33)(24,37,38,48)(26,41,27,39)(29,43,42,47)(31,46,36,44);(9,11)(10,13)(12,15)(14,16)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,33,34,35)(24,29,38,42)(25,39,40,41)(30,44,45,46)(37,47,48,43);(11,15)(13,16)(20,34)(25,40)(30,45)(33,35)(37,48)(39,41)(43,47)(44,46)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6512854);