#9706322

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

4130 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18,22,24)(19,27,28,23)(20,25,30,32)(21,31,29,26)(33,34)(35,43)(36,42)(37,41)(38,44)(39,45)(40,46)(47,48);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19,22,28)(18,23,24,27)(20,29,30,21)(25,31,32,26)(33,35,40,37)(34,41,46,43)(36,45,47,38)(39,42,44,48);(2,7)(3,10)(8,15)(11,12)(18,24)(23,27)(25,32)(26,31)(33,36,40,47)(34,42,46,48)(35,45,37,38)(39,41,44,43);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,31)(30,32)(33,37,40,35)(34,43,46,41)(36,38,47,45)(39,48,44,42);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,16)(10,12)(17,20,22,30)(18,25,24,32)(19,21,28,29)(23,26,27,31)(33,38)(34,44)(35,36)(37,47)(39,46)(40,45)(41,48)(42,43);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,21)(18,26)(19,20)(22,29)(23,25)(24,31)(27,32)(28,30)(33,39,40,44)(34,45,46,38)(35,42,37,48)(36,41,47,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=gh,e^2=hi,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, 9613518);