#9214725

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

4025 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9121921);