#9007113

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

4280 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=gh,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, 8914309);