#8932150

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

3641 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:

(1,2,5,8)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,18)(19,27)(20,26)(21,25)(22,23)(24,28)(29,31)(30,32)(35,43)(39,44)(45,48)(46,57)(53,55)(54,58)(59,61)(63,64);(3,9)(6,13)(10,15)(14,16)(18,23)(19,28)(25,26)(29,30)(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);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,19)(18,24)(20,29)(21,30)(22,28)(23,27)(25,31)(26,32)(33,34)(35,44)(36,40)(37,51)(38,42)(39,43)(41,52)(45,55)(46,54)(47,50)(48,53)(49,60)(56,62)(57,58)(59,63)(61,64);(17,20,22,21)(18,25,23,26)(19,29,28,30)(24,31,27,32)(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);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,28)(29,31)(30,32)(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);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,21,22,20)(18,26,23,25)(19,30,28,29)(24,32,27,31)(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=gh,b^2=gi,d^2=g,e^2=gi,f^2=h,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, 8839346);