#9054443

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

3645 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=gh,e^2=ghi,f^2=h,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, 8961639);