#9203730

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

4248 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)(4,5)(6,9)(8,10)(11,12)(13,16)(14,15)(17,18)(19,26)(20,25)(21,31)(22,24)(23,29)(27,32)(28,30)(33,34,39,38)(35,43,46,40)(36,41,45,47)(37,42,44,48);(1,3)(2,7)(4,11)(5,12)(6,10)(8,9)(13,14)(15,16)(17,19,22,29)(18,23,24,26)(20,27,30,31)(21,28,32,25)(33,35)(34,40)(36,44)(37,45)(38,43)(39,46)(41,42)(47,48);(3,10)(7,8)(11,13)(12,16)(18,24)(20,30)(23,26)(27,31)(33,36,39,45)(34,41,38,47)(35,44,46,37)(40,42,43,48);(1,3)(2,8)(4,13)(5,12)(6,10)(7,9)(11,14)(15,16)(17,20,22,30)(18,25,24,28)(19,27,29,31)(21,26,32,23)(33,36)(34,41)(35,44)(37,46)(38,47)(39,45)(40,42)(43,48);(1,4,6,14)(2,5,9,15)(3,11,10,13)(7,12,8,16)(17,19)(18,26)(20,31)(21,25)(22,29)(23,24)(27,30)(28,32)(33,37,39,44)(34,42,38,48)(35,45,46,36)(40,41,43,47);(1,5)(2,4)(3,12)(6,15)(7,11)(8,13)(9,14)(10,16)(17,21)(18,27)(19,28)(20,26)(22,32)(23,30)(24,31)(25,29)(33,38,39,34)(35,43,46,40)(36,47,45,41)(37,42,44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=h,e^2=ghi,f^2=gh,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, 9110926);