#9053970

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

3718 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8961166);