#6559920

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,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, 6467116);