#8115415

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

3676 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8022611);