#6937210

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1277 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)(25,26,31,44,30,36,29,35)(27,32,45,55,40,49,39,48)(28,34,41,33,43,50,42,51)(37,52,53,46,38,47,54,56);(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)(26,32)(28,37)(29,39)(30,40)(31,45)(33,46)(34,52)(35,48)(36,49)(38,43)(41,53)(42,54)(44,55)(47,50)(51,56);(2,5)(3,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(25,28)(26,33)(27,37)(29,41)(30,43)(31,42)(32,46)(34,44)(35,50)(36,51)(38,40)(39,53)(45,54)(47,48)(49,56)(52,55);(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)(27,38)(29,42)(30,43)(31,41)(32,47)(33,44)(35,51)(36,50)(37,40)(39,53)(45,54)(46,48)(49,52)(55,56);(2,5)(6,8)(10,14)(15,21)(16,22)(23,24)(25,29,30,31)(26,35,36,44)(27,39,40,45)(28,41,43,42)(32,48,49,55)(33,50,51,34)(37,53,38,54)(46,47,56,52)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=f,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,d^c=dfh,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, 6844406);