#9712471

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

4151 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)(17,18,22,24)(19,23,26,21)(20,25,30,32)(27,29,28,31)(33,34)(35,43)(36,40)(37,47)(38,42)(39,44)(41,45)(46,48);(1,3,7,6)(2,8,9,16)(4,11,13,14)(5,15,10,12)(17,19,22,26)(18,23,24,21)(20,28,30,27)(25,31,32,29)(33,35,38,44)(34,39,42,43)(36,45,46,37)(40,47,48,41);(2,9)(4,13)(8,16)(11,14)(18,24)(19,26)(25,32)(27,28)(33,34,38,42)(35,39,44,43)(36,40,46,48)(37,41,45,47);(1,4)(2,10)(3,11)(5,9)(6,14)(7,13)(8,12)(15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,29)(28,31)(30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,44)(45,47)(46,48);(1,5,7,10)(2,4,9,13)(3,12,6,15)(8,14,16,11)(17,20,22,30)(18,25,24,32)(19,27,26,28)(21,31,23,29)(33,36,38,46)(34,40,42,48)(35,37,44,45)(39,41,43,47);(1,6,7,3)(2,8,9,16)(4,14,13,11)(5,15,10,12)(17,21,22,23)(18,26,24,19)(20,29,30,31)(25,27,32,28)(33,37)(34,41)(35,36)(38,45)(39,40)(42,47)(43,48)(44,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,e^2=h,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, 9619667);