#6868771

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1248 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6775967);