#6789568

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1174 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)(25,26,31,35)(27,32,30,45)(28,38,41,55)(29,34,44,51)(33,36,48,40)(37,46,42,49)(39,54,43,56)(47,52,50,53);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,18)(21,24)(25,27,31,30)(26,32,35,45)(28,36,41,40)(29,42,44,37)(33,38,48,55)(34,49,51,46)(39,53,43,52)(47,56,50,54);(2,5)(6,8)(25,28,27,36,31,41,30,40)(26,33,45,55,35,48,32,38)(29,43,37,53,44,39,42,52)(34,50,49,54,51,47,46,56);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,29)(26,34)(27,37)(28,39)(30,42)(31,44)(32,46)(33,47)(35,51)(36,52)(38,54)(40,53)(41,43)(45,49)(48,50)(55,56);(2,5)(6,8)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(29,44)(34,51)(37,42)(39,43)(46,49)(47,50)(52,53)(54,56)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=fh,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,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6696764);