#8828658

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

4268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8735854);