#6565226

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1342 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,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, 6472422);