#6937430

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1313 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)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,31,32,30,35,27,36)(28,39,37,48,42,45,40,33)(29,34,44,46,43,49,38,50)(41,52,51,56,54,55,53,47);(1,2)(3,6)(4,5)(7,8)(10,14)(12,19)(15,20)(18,23)(25,27,30,31)(26,32,35,36)(28,40,42,37)(29,38,43,44)(33,39,48,45)(34,46,49,50)(41,53,54,51)(47,52,56,55);(2,5)(3,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,30,42)(26,33,35,48)(27,37,31,40)(29,41,43,54)(32,45,36,39)(34,47,49,56)(38,51,44,53)(46,55,50,52);(1,3,4,7)(2,6,5,8)(10,14)(11,17)(16,21)(18,23)(25,29,30,43)(26,34,35,49)(27,38,31,44)(28,41,42,54)(32,46,36,50)(33,47,48,56)(37,51,40,53)(39,52,45,55);(2,5)(6,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(29,43)(34,49)(38,44)(41,54)(46,50)(47,56)(51,53)(52,55)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,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, 6844626);