#8993514

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

4122 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8900710);