#9203668

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

4012 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)(3,7)(4,5)(6,9)(8,10)(11,12)(13,16)(14,15)(17,18,23,22)(19,26,21,24)(20,25,28,29)(27,32,30,31)(33,34)(35,36)(37,38)(39,43)(40,41)(42,48)(44,45)(46,47);(1,3)(2,7)(4,11)(5,12)(6,10)(8,9)(13,14)(15,16)(17,19,23,21)(18,24,22,26)(20,27,28,30)(25,31,29,32)(33,35,39,41)(34,40,43,36)(37,46,42,44)(38,45,48,47);(3,10)(7,8)(11,13)(12,16)(20,28)(25,29)(27,30)(31,32)(33,36,39,40)(34,41,43,35)(37,45,42,47)(38,46,48,44);(1,3)(2,8)(4,13)(5,12)(6,10)(7,9)(11,14)(15,16)(17,20,23,28)(18,25,22,29)(19,27,21,30)(24,31,26,32)(33,36)(34,35)(37,47)(38,46)(39,40)(41,43)(42,45)(44,48);(1,4,6,14)(2,5,9,15)(3,11,10,13)(7,12,8,16)(17,21,23,19)(18,24,22,26)(20,27,28,30)(25,32,29,31)(33,37)(34,38)(35,44)(36,45)(39,42)(40,47)(41,46)(43,48);(1,5)(2,4)(3,12)(6,15)(7,11)(8,13)(9,14)(10,16)(17,22,23,18)(19,26,21,24)(20,29,28,25)(27,32,30,31)(33,38)(34,42)(35,45)(36,46)(37,43)(39,48)(40,44)(41,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=gh,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, 9110864);