#6802984

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1242 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)(12,16)(13,18)(14,17)(15,21)(20,24)(22,23)(25,26,29,34,30,35,42,32)(27,31,37,45,38,46,53,43)(28,39,41,47,40,33,48,49)(36,50,52,54,51,44,55,56);(1,2,4,5)(3,7,8,6)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24)(25,27,30,38)(26,31,35,46)(28,36,40,51)(29,37,42,53)(32,43,34,45)(33,44,39,50)(41,52,48,55)(47,54,49,56);(2,5)(6,7)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(26,32)(29,42)(31,43)(33,47)(34,35)(37,53)(39,49)(41,48)(44,54)(45,46)(50,56)(52,55);(10,17)(15,19)(16,18)(23,24)(25,28)(26,33)(27,36)(29,41)(30,40)(31,44)(32,47)(34,49)(35,39)(37,52)(38,51)(42,48)(43,54)(45,56)(46,50)(53,55);(1,3,4,8)(2,6,5,7)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(28,40)(33,39)(36,51)(41,48)(44,50)(47,49)(52,55)(54,56)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6710180);