#9015524

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

4237 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=hi,e^2=i,f^2=h,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, 8922720);