#147688

order	512
nilpotency class	4
isoclinism	isoclinic class of #136630
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1875
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10969
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1005 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2)(3,13,16,44)(4,18,22,9)(5,24)(6,28)(7,11)(8,15,33,49)(10,26)(12,30)(14,42)(17,38,29,43)(19,53)(20,55,58,60)(21,36,23,37)(25,40,50,32)(27,41)(31,56)(34,54)(35,59,52,57)(39,48)(45,61)(46,63)(47,62)(51,64);(1,3)(2,8)(4,19)(5,25)(6,29)(7,16)(9,34)(10,38)(11,33)(12,40)(13,41)(14,21)(15,48)(17,30)(18,51)(20,56)(22,47)(23,46)(24,44)(26,49)(27,50)(28,32)(31,36)(35,63)(37,61)(39,43)(42,52)(45,58)(53,60)(54,57)(55,62)(59,64);(1,4)(2,9)(3,14)(5,20)(6,23)(7,22)(8,31)(10,35)(11,18)(12,37)(13,42)(15,45)(16,46)(17,47)(19,29)(21,30)(24,60)(25,54)(26,59)(27,58)(28,36)(32,51)(33,61)(34,40)(38,62)(39,52)(41,55)(43,53)(44,63)(48,57)(49,56)(50,64)

Polycyclic presentation:

2^9|a^2=g,d^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi

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, 54884);