#8621845

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

4120 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=gh,e^2=ghi,f^2=g,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, 8529041);