#9159954

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

4182 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)(3,10)(4,7)(5,9)(6,8)(11,16)(12,14)(13,15)(17,18)(19,23)(20,25)(21,28)(22,24)(26,32)(27,30)(29,31)(33,34,39,37)(35,43,38,40)(36,41,46,42)(44,48,45,47);(1,3,7,6)(2,8,4,10)(5,13,14,11)(9,16,12,15)(17,19,22,27)(18,23,24,30)(20,26,31,28)(21,25,32,29)(33,35,39,38)(34,40,37,43)(36,44,46,45)(41,47,42,48);(2,4)(8,10)(9,12)(15,16)(18,24)(19,27)(25,29)(26,28)(33,36,39,46)(34,41,37,42)(35,44,38,45)(40,47,43,48);(1,4,7,2)(3,10,6,8)(5,9,14,12)(11,15,13,16)(17,18)(19,23)(20,29)(21,26)(22,24)(25,31)(27,30)(28,32)(33,36)(34,42)(35,44)(37,41)(38,45)(39,46)(40,48)(43,47);(1,5,7,14)(2,9,4,12)(3,11,6,13)(8,15,10,16)(17,20)(18,25)(19,28)(21,23)(22,31)(24,29)(26,27)(30,32)(33,37,39,34)(35,43,38,40)(36,42,46,41)(44,48,45,47);(1,6,7,3)(2,10,4,8)(5,11,14,13)(9,15,12,16)(17,21)(18,26)(19,25)(20,30)(22,32)(23,31)(24,28)(27,29)(33,38,39,35)(34,40,37,43)(36,45,46,44)(41,47,42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=ghi,d^2=hi,e^2=g,f^2=g,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, 9067150);