#500756

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
prime radical	#61106	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1006 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)(3,17)(4,23)(5,9)(6,15)(7,33)(8,13)(10,16)(11,19)(12,26)(14,35)(18,42)(20,57)(21,50)(22,48)(24,52)(25,46)(27,43)(28,44)(29,47)(30,61)(31,32)(34,40)(36,39)(37,41)(38,60)(45,51)(49,54)(53,62)(55,64)(56,63)(58,59);(1,3,6,19,10,22,9,37)(2,11,5,17,16,41,15,48)(4,24,25,59,29,60,28,54)(7,34,32,20,35,21,8,36)(12,42,43,63,45,64,30,62)(13,40,14,39,31,50,33,57)(18,51,53,27,55,26,56,61)(23,58,44,52,47,49,46,38);(1,4,10,29)(2,12,16,45)(3,18,22,55)(5,30,15,43)(6,28,9,25)(7,27,35,61)(8,26,32,51)(11,38,41,52)(13,46,31,44)(14,47,33,23)(17,49,48,58)(19,56,37,53)(20,24,36,60)(21,54,34,59)(39,64,57,42)(40,62,50,63);(1,5,9,2,10,15,6,16)(3,17,37,11,22,48,19,41)(4,12,28,30,29,45,25,43)(7,14,8,13,35,33,32,31)(18,52,56,58,55,38,53,49)(20,50,34,39,36,40,21,57)(23,27,46,51,47,61,44,26)(24,42,54,62,60,64,59,63)

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=ei,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, 407952);