#6774561

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1189 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fi,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,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, 6681757);