#9726459

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

4137 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=gh,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,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, 9633655);