#9224348

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	#49658	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4144 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9131544);