#6755867

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1367 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)(11,19)(12,16)(13,18)(14,17)(15,21)(20,24)(22,23)(25,26,30,35)(27,37,40,32)(28,41,44,52)(29,36,31,34)(33,46,50,43)(38,53,42,56)(39,48,45,51)(47,55,49,54);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,17,19)(12,20,13,22)(16,23,18,24)(25,27,30,40)(26,32,35,37)(28,42,44,38)(29,45,31,39)(33,49,50,47)(34,51,36,48)(41,53,52,56)(43,54,46,55);(1,3,4,8)(2,5,6,7)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,28,30,44)(26,33,35,50)(27,38,40,42)(29,43,31,46)(32,47,37,49)(34,41,36,52)(39,54,45,55)(48,53,51,56);(10,17)(15,19)(16,18)(23,24)(25,29,30,31)(26,34,35,36)(27,39,40,45)(28,43,44,46)(32,48,37,51)(33,41,50,52)(38,54,42,55)(47,53,49,56);(2,6)(5,7)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(28,44)(33,50)(38,42)(41,52)(43,46)(47,49)(53,56)(54,55)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fhi,d^2=i,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,e^c=ei,g^c=gi

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, 6663063);