#9910676

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3748 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=gh,e^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, 9817872);