#6937198

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fi,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, 6844394);