#6804862

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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

1187 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=fg,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6712058);