#9685547

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3897 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,23,27)(19,26,21,24)(20,29,22,25)(28,31,30,32)(33,34)(35,42)(36,40)(37,47)(38,43)(39,46)(41,44)(45,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,23,21)(18,24,27,26)(20,28,22,30)(25,31,29,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,47)(41,48)(42,43);(2,3)(7,9)(8,10)(14,15)(17,20,23,22)(18,25,27,29)(19,28,21,30)(24,31,26,32)(33,36,38,45)(34,40,43,48)(35,44,46,37)(39,47,42,41);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,21,23,19)(18,26,27,24)(20,28,22,30)(25,31,29,32)(33,37,38,44)(34,41,43,47)(35,45,46,36)(39,48,42,40);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,22,23,20)(18,25,27,29)(19,28,21,30)(24,32,26,31)(33,35)(34,42)(36,44)(37,45)(38,46)(39,43)(40,41)(47,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,18)(19,24)(20,25)(21,26)(22,29)(23,27)(28,31)(30,32)(33,34)(35,39)(36,40)(37,41)(38,43)(42,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=g,e^2=h,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, 9592743);