#8093500

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3711 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=i,e^2=h,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, 8000696);