#8772474

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

4146 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8679670);