#8720393

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

4119 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)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18)(19,24)(20,29)(21,26)(22,23)(25,30)(27,28)(31,32)(33,34)(35,43)(36,40)(37,39)(38,42)(41,46)(44,48)(45,47);(2,3)(7,9)(8,10)(15,16)(17,18)(19,27)(20,25)(21,26)(22,23)(24,28)(29,30)(31,32)(33,35,38,37)(34,39,42,43)(36,44,46,45)(40,47,41,48);(2,3)(7,9)(8,10)(15,16)(18,23)(19,28)(25,29)(26,32)(33,36,38,46)(34,40,42,41)(35,44,37,45)(39,47,43,48);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,19)(18,24)(20,26)(21,29)(22,28)(23,27)(25,31)(30,32)(33,36)(34,41)(35,44)(37,45)(38,46)(39,48)(40,42)(43,47);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20)(18,25)(19,26)(21,27)(22,30)(23,29)(24,31)(28,32)(33,37,38,35)(34,39,42,43)(36,45,46,44)(40,47,41,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21,22,31)(18,26,23,32)(19,25,28,29)(20,24,30,27)(33,34)(35,39)(36,40)(37,43)(38,42)(41,46)(44,47)(45,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8627589);