#9481516

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	#392	C₂³
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)(3,5)(4,8)(6,10)(7,9)(11,13)(12,14)(15,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,34)(35,40)(36,41)(37,42)(38,39)(43,53)(44,54)(45,51)(46,48)(47,55)(49,56)(50,52)(57,59)(58,63)(60,62)(61,64);(1,3,6,9)(2,7,10,5)(4,11,14,16)(8,15,12,13)(18,19)(23,25)(24,26)(30,31)(34,39)(40,51)(41,46)(42,52)(53,57)(54,56)(55,60)(63,64);(1,2,6,10)(3,7,9,5)(4,8,14,12)(11,15,16,13)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,35,38,45)(34,40,39,51)(36,43,48,59)(37,49,50,44)(41,53,46,57)(42,56,52,54)(47,61,62,58)(55,64,60,63);(1,2)(3,7)(4,12)(5,9)(6,10)(8,14)(11,13)(15,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36)(34,41)(35,43)(37,47)(38,48)(39,46)(40,53)(42,55)(44,58)(45,59)(49,61)(50,62)(51,57)(52,60)(54,63)(56,64);(1,4,6,14)(2,8,10,12)(3,11,9,16)(5,13,7,15)(33,34)(35,40)(36,46)(37,42)(38,39)(41,48)(43,57)(44,54)(45,51)(47,60)(49,56)(50,52)(53,59)(55,62)(58,64)(61,63);(1,5,6,7)(2,9,10,3)(4,13,14,15)(8,16,12,11)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37,38,50)(34,42,39,52)(35,44,45,49)(36,47,48,62)(40,54,51,56)(41,55,46,60)(43,58,59,61)(53,63,57,64)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=i,e^2=h,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^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, 9388712);