#9481445

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

3655 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=hi,e^2=g,f^2=h,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, 9388641);