#6915202

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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

1378 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)(11,19)(12,13)(14,17)(15,18)(16,22)(20,23)(21,24)(25,26,30,35)(27,36,31,32)(28,39,41,47)(29,43,45,34)(33,37,50,42)(38,48,46,52)(40,56,44,55)(49,54,51,53);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,24)(18,19)(25,27,30,31)(26,32,35,36)(28,37,41,42)(29,38,45,46)(33,47,50,39)(34,48,43,52)(40,53,44,54)(49,56,51,55);(1,3)(2,5)(4,8)(6,7)(9,12,15,17)(10,13,18,14)(11,20,22,24)(16,21,19,23)(25,28,27,37,30,41,31,42)(26,33,32,47,35,50,36,39)(29,44,38,54,45,40,46,53)(34,51,48,55,43,49,52,56);(3,8)(5,7)(9,13,15,14)(10,17,18,12)(11,21,22,23)(16,24,19,20)(25,29,30,45)(26,34,35,43)(27,38,31,46)(28,40,41,44)(32,48,36,52)(33,49,50,51)(37,53,42,54)(39,55,47,56);(2,6)(5,7)(9,14,15,13)(10,17,18,12)(11,21,22,23)(16,20,19,24)(29,45)(34,43)(38,46)(40,44)(48,52)(49,51)(53,54)(55,56)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6822398);