#8634845

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	#35109	C₂
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)(3,8)(4,9)(5,7)(6,10)(11,14)(12,16)(13,15)(17,18,23,22)(19,24,26,21)(20,25,29,30)(27,31,32,28)(33,34,39,37)(35,40,36,43)(38,41,46,45)(42,47,44,48);(1,3,7,10)(2,8,5,6)(4,12,15,11)(9,16,13,14)(17,19)(18,24)(20,27)(21,22)(23,26)(25,31)(28,30)(29,32)(34,37)(40,43)(41,45)(47,48);(2,5)(3,10)(9,13)(11,12)(19,26)(21,24)(27,32)(28,31)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(1,4,7,15)(2,9,5,13)(3,11,10,12)(6,16,8,14)(17,20)(18,25)(19,27)(21,28)(22,30)(23,29)(24,31)(26,32)(33,36,39,35)(34,40,37,43)(38,42,46,44)(41,48,45,47);(1,5,7,2)(3,8,10,6)(4,13,15,9)(11,14,12,16)(17,21)(18,19)(20,28)(22,26)(23,24)(25,27)(29,31)(30,32)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,6,7,8)(2,10,5,3)(4,14,15,16)(9,11,13,12)(17,22,23,18)(19,21,26,24)(20,25,29,30)(27,31,32,28)(33,38,39,46)(34,41,37,45)(35,42,36,44)(40,47,43,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8542041);