#8932091

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

3709 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)(18,23)(21,29)(24,30)(28,32)(33,34,39,37)(35,40,46,44)(36,41,49,50)(38,42,53,51)(43,54,47,56)(45,55,52,58)(48,57,61,62)(59,63,60,64);(1,3,6,11)(2,7,9,4)(5,10,14,16)(8,15,13,12)(20,27)(24,30)(26,31)(28,32)(34,37)(40,44)(41,50)(42,51)(54,56)(55,58)(57,62)(63,64);(1,2)(3,4)(5,8)(6,9)(7,11)(10,12)(13,14)(15,16)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,35,39,46)(34,40,37,44)(36,47,49,43)(38,52,53,45)(41,56,50,54)(42,58,51,55)(48,59,61,60)(57,63,62,64);(1,4)(2,3)(5,12)(6,7)(8,10)(9,11)(13,16)(14,15)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32)(33,36,39,49)(34,41,37,50)(35,43,46,47)(38,48,53,61)(40,54,44,56)(42,57,51,62)(45,59,52,60)(55,63,58,64);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,15)(14,16)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,37,39,34)(35,44,46,40)(36,41,49,50)(38,51,53,42)(43,54,47,56)(45,58,52,55)(48,57,61,62)(59,63,60,64);(1,5,6,14)(2,8,9,13)(3,10,11,16)(4,12,7,15)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,38,39,53)(34,42,37,51)(35,45,46,52)(36,48,49,61)(40,55,44,58)(41,57,50,62)(43,59,47,60)(54,63,56,64)

Polycyclic presentation:

2^9|a^2=g,b^2=i,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, 8839287);