#6820536

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1272 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=fgh,d^2=fi,e^2=hi,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, 6727732);