#6937526

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1311 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)(25,26,30,34,29,33,42,48)(27,35,38,51,37,31,44,46)(28,39,41,47,40,49,43,32)(36,52,54,45,53,56,55,50);(1,2)(3,6)(4,5)(7,8)(10,14)(12,19)(15,20)(18,23)(25,27,29,37)(26,31,33,35)(28,36,40,53)(30,38,42,44)(32,45,47,50)(34,46,48,51)(39,56,49,52)(41,54,43,55);(2,5)(3,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,29,40)(26,32,33,47)(27,36,37,53)(30,43,42,41)(31,45,35,50)(34,49,48,39)(38,55,44,54)(46,52,51,56);(1,3,4,7)(2,6,5,8)(10,14)(11,17)(16,21)(18,23)(26,33)(27,37)(28,40)(34,48)(38,44)(41,43)(45,50)(52,56);(2,5)(6,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(26,33)(27,37)(32,47)(34,48)(36,53)(38,44)(39,49)(54,55)

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=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi

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, 6844722);