#9706327

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18,22,24)(19,20,27,23)(21,29,31,25)(26,32,30,28)(34,36)(40,41)(42,44)(47,48);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19,22,27)(18,23,24,20)(21,26,31,30)(25,32,29,28)(33,34,39,36)(35,40,37,41)(38,42,46,44)(43,47,45,48);(2,7)(3,10)(8,15)(11,12)(18,24)(20,23)(25,29)(28,32)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,20)(18,19)(21,28)(22,23)(24,27)(25,26)(29,30)(31,32)(33,36,39,34)(35,41,37,40)(38,44,46,42)(43,48,45,47);(1,4,6,14)(2,8,7,15)(3,11,10,12)(5,13,9,16)(17,19)(18,20)(21,30)(22,27)(23,24)(25,32)(26,31)(28,29)(33,37,39,35)(34,41,36,40)(38,43,46,45)(42,47,44,48);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,21)(18,25)(19,26)(20,28)(22,31)(23,32)(24,29)(27,30)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9613523);