#9379899

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

4364 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)(19,23)(20,24)(21,30)(22,26)(25,31)(27,29)(28,32)(34,38)(36,44)(39,41)(42,48);(2,5)(7,9)(8,12)(15,16)(17,19,22,27)(18,23,26,29)(20,21,28,31)(24,30,32,25)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(3,10)(7,9)(11,13)(15,16)(19,27)(20,28)(23,29)(24,32)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,3,6,10)(2,7,5,9)(4,13,14,11)(8,16,12,15)(17,20)(18,24)(19,21)(22,28)(23,30)(25,29)(26,32)(27,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4,6,14)(2,8,5,12)(3,11,10,13)(7,15,9,16)(17,21)(18,25)(19,28)(20,27)(22,31)(23,24)(26,30)(29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5,6,2)(3,9,10,7)(4,12,14,8)(11,16,13,15)(17,18)(19,29)(20,24)(21,25)(22,26)(23,27)(28,32)(30,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=gh,e^2=gh,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=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9287095);