#8340002

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

4151 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8247198);