#6891462

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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1378 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)(11,16)(12,21)(13,17)(14,18)(15,20)(19,23)(22,24)(25,26,31,36)(27,32,39,50)(28,34,42,52)(29,35,30,45)(33,44,53,41)(37,47,40,54)(38,48,43,49)(46,55,51,56);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,18)(17,23)(21,24)(25,27,31,39)(26,32,36,50)(28,40,42,37)(29,43,30,38)(33,51,53,46)(34,54,52,47)(35,49,45,48)(41,55,44,56);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,19)(14,20)(15,18)(17,23)(21,24)(25,28,31,42)(26,33,36,53)(27,37,39,40)(29,41,30,44)(32,46,50,51)(34,35,52,45)(38,55,43,56)(47,49,54,48);(3,8)(5,7)(9,12,14,13)(10,17,18,21)(11,19,20,22)(15,23,16,24)(25,28)(26,34)(27,37)(29,44)(30,41)(31,42)(32,47)(33,45)(35,53)(36,52)(38,56)(39,40)(43,55)(46,48)(49,51)(50,54);(2,6)(5,7)(9,13,14,12)(10,17,18,21)(11,19,20,22)(15,24,16,23)(25,29,31,30)(26,35,36,45)(27,38,39,43)(28,41,42,44)(32,48,50,49)(33,52,53,34)(37,55,40,56)(46,47,51,54)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,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, 6798658);