#6565557

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1277 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)(3,7)(4,8)(5,6)(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)(32,39)(33,42)(37,54)(38,49)(41,53)(43,52)(44,48)(45,51)(47,55)(50,56);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,44,31,38,30,39)(26,32,46,49,35,48,34,36)(28,41,45,56,42,37,43,55)(33,50,51,53,40,47,52,54);(2,6)(7,8)(11,17)(14,20)(18,23)(21,24)(25,28)(26,33)(27,37)(29,45)(30,43)(31,42)(32,47)(34,52)(35,40)(36,53)(38,41)(39,56)(44,55)(46,51)(48,50)(49,54);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(27,38)(28,42)(32,48)(33,40)(36,49)(39,44)(43,45)(51,52);(1,4,5,3)(2,8,6,7)(12,19)(15,22)(18,23)(21,24)(25,29,31,30)(26,34,35,46)(27,39,38,44)(28,43,42,45)(32,49,48,36)(33,51,40,52)(37,55,41,56)(47,53,50,54)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,d^2=gi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ei,d^c=di,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, 6472753);