#9206498

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

4135 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)(3,8)(4,9)(5,13)(6,7)(10,15)(11,14)(12,16)(17,18)(19,26)(20,28)(21,24)(22,23)(25,30)(27,31)(29,32)(33,34,40,44)(35,45,37,41)(36,42,46,47)(38,39,43,48);(2,7)(4,12)(8,14)(10,13)(19,21)(24,26)(27,29)(31,32)(33,35,40,37)(34,41,44,45)(36,38,46,43)(39,42,48,47);(2,7)(3,11)(9,16)(10,13)(17,19,23,21)(18,24,22,26)(20,29,30,27)(25,32,28,31)(33,36,40,46)(34,42,44,47)(35,38,37,43)(39,41,48,45);(1,3)(2,8)(4,13)(5,9)(6,11)(7,14)(10,12)(15,16)(17,20,23,30)(18,25,22,28)(19,27,21,29)(24,31,26,32)(33,37,40,35)(34,41,44,45)(36,38,46,43)(39,47,48,42);(1,4)(2,9)(3,10)(5,14)(6,12)(7,16)(8,15)(11,13)(17,21,23,19)(18,24,22,26)(20,27,30,29)(25,32,28,31)(33,38)(34,39)(35,46)(36,37)(40,43)(41,42)(44,48)(45,47);(1,5,6,15)(2,10,7,13)(3,9,11,16)(4,8,12,14)(17,22,23,18)(19,26,21,24)(20,28,30,25)(27,32,29,31)(33,39)(34,43)(35,42)(36,41)(37,47)(38,44)(40,48)(45,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=gh,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9113694);