#9039128

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3707 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=i,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

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, 8946324);