#9166431

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	#43080	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4244 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)(3,7)(4,8)(5,14)(6,9)(10,12)(11,15)(13,16)(17,18)(19,28)(20,25)(21,22)(23,27)(24,31)(26,30)(29,32)(33,34,40,37)(35,39,43,45)(36,41,38,42)(44,47,46,48);(1,3,6,13)(2,7,9,16)(4,11,14,10)(5,12,8,15)(17,19)(18,24)(20,29)(21,30)(22,25)(23,31)(26,32)(27,28)(33,35)(34,39)(36,46)(37,45)(38,44)(40,43)(41,48)(42,47);(1,4,6,14)(2,5,9,8)(3,10,13,11)(7,15,16,12)(17,20,23,30)(18,25,27,26)(19,29,31,21)(22,28,32,24)(33,36,40,38)(34,41,37,42)(35,44,43,46)(39,47,45,48);(1,4)(2,8)(3,11)(5,9)(6,14)(7,15)(10,13)(12,16)(17,20)(18,26)(19,29)(21,31)(22,28)(23,30)(24,32)(25,27)(33,37,40,34)(35,45,43,39)(36,41,38,42)(44,47,46,48);(1,3)(2,7)(4,10)(5,15)(6,13)(8,12)(9,16)(11,14)(17,21,23,29)(18,22,27,32)(19,30,31,20)(24,25,28,26)(33,38,40,36)(34,42,37,41)(35,44,43,46)(39,47,45,48);(1,5,6,8)(2,4,9,14)(3,12,13,15)(7,11,16,10)(17,22)(18,21)(19,25)(20,28)(23,32)(24,30)(26,31)(27,29)(33,39,40,45)(34,43,37,35)(36,47,38,48)(41,46,42,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=hi,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9073627);