#9203794

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

4129 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,e^2=gi,f^2=g,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, 9110990);