#6749010

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1391 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,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, 6656206);