#6937201

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1275 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)(25,26,29,34,30,35,42,32)(27,31,37,45,38,46,52,43)(28,33,41,50,40,49,54,47)(36,44,51,56,39,48,53,55);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,15,18,21)(12,16,19,22)(17,23,20,24)(25,27,30,38)(26,31,35,46)(28,39,40,36)(29,37,42,52)(32,43,34,45)(33,48,49,44)(41,53,54,51)(47,56,50,55);(2,5)(3,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(26,32)(29,42)(31,43)(33,47)(34,35)(37,52)(41,54)(44,55)(45,46)(48,56)(49,50)(51,53);(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)(26,33)(27,36)(29,41)(30,40)(31,44)(32,47)(34,50)(35,49)(37,51)(38,39)(42,54)(43,55)(45,56)(46,48)(52,53);(2,5)(6,8)(10,14)(15,21)(16,22)(23,24)(28,40)(33,49)(36,39)(41,54)(44,48)(47,50)(51,53)(55,56)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=f,d^2=fh,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, 6844397);