#6565426

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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

1319 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26)(27,36)(28,40)(29,34)(30,46)(31,35)(32,38)(33,43)(37,54)(39,48)(41,53)(42,52)(44,49)(45,51)(47,55)(50,56);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,38,31,39,29,44)(26,32,34,36,35,49,46,48)(28,41,42,55,43,37,45,56)(33,50,52,54,40,47,51,53);(2,5)(6,8)(11,17)(14,20)(18,23)(21,24)(25,28)(26,33)(27,37)(29,45)(30,42)(31,43)(32,47)(34,52)(35,40)(36,53)(38,56)(39,41)(44,55)(46,51)(48,54)(49,50);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24);(12,19)(15,22)(18,23)(21,24)(25,29,31,30)(26,34,35,46)(27,38,39,44)(28,42,43,45)(32,48,49,36)(33,51,40,52)(37,55,41,56)(47,53,50,54)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,e^c=ei,e^d=ehi

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, 6472622);