#6519059

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499792
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

1688 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)(17,18,22,26)(19,27,29,42)(20,24,32,40)(21,34,36,25)(23,35,38,30)(28,43,31,45)(33,48,47,41)(37,46,39,44);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,31)(21,35)(22,29)(24,39)(25,42)(26,38)(27,34)(28,32)(30,36)(33,44)(37,40)(41,43)(45,48)(46,47);(2,6)(5,8)(10,13)(11,15)(17,20)(18,24)(19,28)(21,33)(22,32)(23,37)(25,41)(26,40)(27,43)(29,31)(30,44)(34,48)(35,46)(36,47)(38,39)(42,45);(9,11,12,15)(10,14,13,16)(20,32)(24,40)(28,31)(33,47)(37,39)(41,48)(43,45)(44,46);(10,13)(14,16)(19,29)(20,32)(23,38)(24,40)(27,42)(30,35)(33,47)(41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei

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, 6426255);