#8339963

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

4133 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)(5,8,13,16)(11,15,12,14)(17,18)(19,24)(20,23)(21,25)(22,26)(27,31)(28,30)(29,32)(33,34)(35,39)(36,43)(37,42)(38,48)(40,45)(41,47)(44,46);(3,4)(7,10)(11,12)(14,15)(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(33,35)(34,41)(36,46)(37,38)(39,45)(40,47)(42,44)(43,48);(1,3,6,4)(2,7,9,10)(5,12,13,11)(8,15,16,14)(17,19,22,20)(18,24,26,23)(21,27,29,28)(25,31,32,30)(33,36,40,37)(34,42,45,43)(35,46,47,38)(39,48,41,44);(1,4,6,3)(2,7,9,10)(5,12,13,11)(8,14,16,15)(17,20,22,19)(18,24,26,23)(21,27,29,28)(25,30,32,31)(33,37,40,36)(34,43,45,42)(35,46,47,38)(39,48,41,44);(1,2)(3,7)(4,10)(5,8)(6,9)(11,14)(12,15)(13,16)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,38,40,46)(34,44,45,48)(35,37,47,36)(39,43,41,42);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,16)(10,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,39,40,41)(34,35,45,47)(36,44,37,48)(38,43,46,42)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gi,d^2=gi,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, 8247159);