#10247015

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3707 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:

(1,2,8,13)(3,7,12,9)(4,10,5,14)(6,11,15,16)(34,39)(37,50)(40,44)(41,52)(46,59)(49,51)(54,56)(61,62);(1,3,8,12)(2,9,13,7)(4,6,5,15)(10,16,14,11)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,29,31)(34,39)(35,43)(36,47)(42,53)(46,59)(49,51)(54,56)(63,64);(1,4,8,5)(2,10,13,14)(3,6,12,15)(7,11,9,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(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);(1,5,8,4)(2,10,13,14)(3,6,12,15)(7,16,9,11)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,32,29)(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);(1,6)(2,11)(3,5)(4,12)(7,14)(8,15)(9,10)(13,16)(17,19)(18,20)(21,29)(22,27)(23,25)(24,31)(26,30)(28,32)(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);(1,7,8,9)(2,12,13,3)(4,11,5,16)(6,10,15,14)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(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|a^2=g,b^2=ghi,c^2=gh,d^2=gi,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, 10154211);