#9159923

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
prime radical	#59992	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4124 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)(3,12)(4,14)(5,11)(6,8)(7,9)(10,15)(13,16)(17,18,22,23)(19,24,27,30)(20,25,31,28)(21,29,32,26)(34,36)(40,41)(42,44)(47,48);(1,3,7,6)(2,8,9,12)(4,13,15,11)(5,10,16,14)(17,18,22,23)(19,24,27,30)(20,28,31,25)(21,26,32,29)(33,34,39,36)(35,40,37,41)(38,42,46,44)(43,47,45,48);(2,9)(4,15)(8,12)(11,13)(18,23)(19,27)(25,28)(26,29)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,4)(2,10)(3,13)(5,12)(6,11)(7,15)(8,16)(9,14)(17,19)(18,24)(20,29)(21,28)(22,27)(23,30)(25,32)(26,31)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,5,7,16)(2,11,9,13)(3,14,6,10)(4,8,15,12)(17,20)(18,25)(19,26)(21,24)(22,31)(23,28)(27,29)(30,32)(33,37,39,35)(34,41,36,40)(38,43,46,45)(42,47,44,48);(1,6,7,3)(2,12,9,8)(4,11,15,13)(5,14,16,10)(17,21)(18,26)(19,28)(20,30)(22,32)(23,29)(24,31)(25,27)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=ghi,d^2=ghi,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, 9067119);