#8093616

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3657 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000812);