#10247227

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

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,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, 10154423);