#9910699

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=h,e^2=g,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^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, 9817895);