#6760638

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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

1202 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)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)(25,26,31,33,30,36,29,35)(27,32,42,44,39,47,38,46)(28,34,43,48,40,49,41,50)(37,45,53,54,51,55,52,56);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,18)(21,24)(25,27,30,39)(26,32,36,47)(28,37,40,51)(29,38,31,42)(33,44,35,46)(34,45,49,55)(41,52,43,53)(48,54,50,56);(1,3,4,8)(2,5,6,7)(26,33)(29,31)(32,44)(34,48)(35,36)(38,42)(41,43)(45,54)(46,47)(49,50)(52,53)(55,56);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,28,30,40)(26,34,36,49)(27,37,39,51)(29,41,31,43)(32,45,47,55)(33,48,35,50)(38,52,42,53)(44,54,46,56);(2,6)(5,7)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(28,40)(34,49)(37,51)(41,43)(45,55)(48,50)(52,53)(54,56)

Polycyclic presentation:

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

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, 6667834);