#9923989

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3677 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)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,32)(30,31)(33,34,39,42)(35,43,46,36)(37,40,49,53)(38,51,52,41)(44,55,57,47)(45,48,59,56)(50,62,61,54)(58,60,64,63);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(19,27)(21,30)(23,32)(25,31)(37,49)(40,53)(44,57)(47,55)(50,61)(54,62)(58,64)(60,63);(17,19,22,27)(18,23,26,32)(20,21,28,30)(24,31,29,25)(33,35,39,46)(34,36,42,43)(37,44,49,57)(38,45,52,59)(40,47,53,55)(41,48,51,56)(50,58,61,64)(54,60,62,63);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,20,22,28)(18,24,26,29)(19,21,27,30)(23,31,32,25)(33,36)(34,35)(37,47)(38,48)(39,43)(40,44)(41,45)(42,46)(49,55)(50,60)(51,59)(52,56)(53,57)(54,58)(61,63)(62,64);(1,2)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(17,21)(18,25)(19,28)(20,27)(22,30)(23,24)(26,31)(29,32)(33,37,39,49)(34,40,42,53)(35,44,46,57)(36,47,43,55)(38,50,52,61)(41,54,51,62)(45,58,59,64)(48,60,56,63);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,18)(19,23)(20,24)(21,31)(22,26)(25,30)(27,32)(28,29)(33,38,39,52)(34,41,42,51)(35,45,46,59)(36,48,43,56)(37,50,49,61)(40,54,53,62)(44,58,57,64)(47,60,55,63)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=gi,e^2=h,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^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9831185);