#9713343

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	#37216	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4019 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=h,e^2=i,f^2=ghi,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, 9620539);