#9723166

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

4126 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,e^2=h,f^2=hi,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, 9630362);