#8932135

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	#392	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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839331);