#6937281

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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

1256 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6844477);