#9481534

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

3639 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388730);