#8479517

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

4038 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)(3,9)(4,10)(5,8)(6,7)(11,15)(12,16)(13,14)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,29)(30,31)(36,44)(39,45)(43,47)(46,48);(2,7)(3,10)(8,14)(11,16)(17,19,23,20)(18,24,27,25)(21,30,32,28)(22,29,26,31)(33,34)(35,42)(36,45)(37,40)(38,41)(39,44)(43,46)(47,48);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,19,23,20)(18,25,27,24)(21,28,32,30)(22,29,26,31)(33,35,38,40)(34,37,41,42)(36,43,44,47)(39,46,45,48);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,20,23,19)(18,25,27,24)(21,30,32,28)(22,29,26,31)(33,34)(35,37)(36,39)(38,41)(40,42)(43,46)(44,45)(47,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,21)(18,26)(19,28)(20,30)(22,27)(23,32)(24,29)(25,31)(33,36)(34,39)(35,43)(37,46)(38,44)(40,47)(41,45)(42,48);(1,5,6,13)(2,8,7,14)(3,11,10,16)(4,12,9,15)(17,22,23,26)(18,21,27,32)(19,29,20,31)(24,30,25,28)(33,37)(34,40)(35,41)(36,46)(38,42)(39,47)(43,45)(44,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8386713);