#6745133

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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

1176 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)(27,32)(28,34)(29,35)(30,33)(31,36)(37,50)(38,45)(39,46)(40,44)(41,47)(42,48)(43,49)(51,55)(52,54)(53,56);(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,30,39)(26,32,33,46)(28,38,41,53)(29,40,31,37)(34,45,47,56)(35,44,36,50)(42,52,43,51)(48,54,49,55);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,33)(27,37)(28,41)(29,31)(32,44)(38,52)(39,40)(45,55)(46,50)(48,49)(51,53)(54,56);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,28,30,41)(26,34,33,47)(27,38,39,53)(29,42,31,43)(32,45,46,56)(35,48,36,49)(37,51,40,52)(44,54,50,55);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,29)(26,35)(28,42)(30,31)(33,36)(34,48)(37,40)(41,43)(44,50)(47,49)(51,52)(54,55)

Polycyclic presentation:

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

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, 6652329);