#6846372

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1165 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fhi,d^2=fgi,e^2=fghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6753568);