#6739918

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1180 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=gi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,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, 6647114);