#10247007

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

3712 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)(34,39)(37,50)(40,44)(41,52)(46,59)(49,51)(54,56)(61,62);(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)(34,39)(35,43)(36,47)(42,53)(46,59)(49,51)(54,56)(63,64);(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)(33,34,38,39)(35,44,43,40)(36,41,47,52)(37,42,50,53)(45,56,48,54)(46,60,59,55)(49,57,51,58)(61,64,62,63);(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)(33,35,38,43)(34,40,39,44)(36,48,47,45)(37,46,50,59)(41,56,52,54)(42,55,53,60)(49,62,51,61)(57,64,58,63);(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)(33,36,38,47)(34,41,39,52)(35,45,43,48)(37,51,50,49)(40,54,44,56)(42,58,53,57)(46,62,59,61)(55,64,60,63);(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)(33,37)(34,42)(35,46)(36,49)(38,50)(39,53)(40,55)(41,57)(43,59)(44,60)(45,61)(47,51)(48,62)(52,58)(54,63)(56,64)

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=i,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, 10154203);