#6741164

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=f,d^2=g,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,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, 6648360);