#9159920

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	#41140	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 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:

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

Polycyclic presentation:

2^9|a^2=gi,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, 9067116);