#500753

order	512
nilpotency class	4
isoclinism	isoclinic class of #500753
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24248
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1002 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=e,d^2=e,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fhi,g^b=gh,h^b=hi,d^c=dh,e^c=ei,f^c=fh,g^c=ghi,h^c=hi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407949);