#6612364

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1626 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18,22,26)(19,27,29,41)(20,24,31,38)(21,33,34,25)(23,35,37,30)(28,40,42,48)(32,45,44,39)(36,46,47,43);(9,11)(10,13)(12,16)(14,15)(17,19,21,30,22,29,34,35)(18,23,25,27,26,37,33,41)(20,28,32,43,31,42,44,46)(24,36,39,40,38,47,45,48);(2,5)(6,8)(11,16)(13,14)(17,20)(18,24)(19,28)(21,32)(22,31)(23,36)(25,39)(26,38)(27,40)(29,42)(30,43)(33,45)(34,44)(35,46)(37,47)(41,48);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,16)(13,15)(20,31)(24,38)(28,42)(32,44)(36,47)(39,45)(40,48)(43,46);(11,16)(13,14)(19,29)(20,31)(23,37)(24,38)(27,41)(30,35)(32,44)(39,45)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dhi,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, 6519560);