#6937156

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1254 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)(25,26,31,45,30,36,29,35)(27,37,39,49,40,32,41,48)(28,34,42,33,43,50,44,51)(38,53,56,52,55,47,54,46);(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)(28,38)(29,41)(30,40)(31,39)(33,46)(34,47)(35,49)(36,37)(42,56)(43,55)(44,54)(45,48)(50,53)(51,52);(2,5)(3,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(25,28,30,43)(26,33,36,51)(27,38,40,55)(29,42,31,44)(32,46,37,52)(34,35,50,45)(39,54,41,56)(47,49,53,48);(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)(29,44)(30,43)(31,42)(32,47)(33,45)(35,51)(36,50)(37,53)(39,56)(40,55)(41,54)(46,48)(49,52);(2,5)(6,8)(10,14)(15,21)(16,22)(23,24)(25,29,30,31)(26,35,36,45)(27,39,40,41)(28,42,43,44)(32,48,37,49)(33,50,51,34)(38,54,55,56)(46,47,52,53)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=f,d^2=fhi,e^2=i,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, 6844352);