#9127353

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

4199 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9034549);