#9015378

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

4265 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=ghi,e^2=i,f^2=gh,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, 8922574);