#9206392

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

3995 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=gh,e^2=gi,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, 9113588);