#6818075

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1154 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fgh,d^2=fi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6725271);