#10247188

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

3709 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10154384);