#9545558

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	#48995	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3858 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,23,26)(19,24,21,22)(20,25,29,32)(27,31,30,28)(33,34)(35,39)(36,40)(37,48)(38,42)(41,45)(43,44)(46,47);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,23,21)(18,24,26,22)(20,27,29,30)(25,31,32,28)(33,35,38,44)(34,39,42,43)(36,45,47,37)(40,41,46,48);(2,3)(7,9)(8,10)(14,15)(19,21)(22,24)(27,30)(28,31)(33,34,38,42)(35,43,44,39)(36,46,47,40)(37,48,45,41);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,29)(18,25,26,32)(19,27,21,30)(22,28,24,31)(33,34)(35,39)(36,46)(37,41)(38,42)(40,47)(43,44)(45,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21,23,19)(18,22,26,24)(20,27,29,30)(25,31,32,28)(33,36)(34,40)(35,37)(38,47)(39,48)(41,43)(42,46)(44,45);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22)(18,19)(20,28)(21,26)(23,24)(25,27)(29,31)(30,32)(33,37)(34,41)(35,36)(38,45)(39,46)(40,43)(42,48)(44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,d^2=ghi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9452754);