#9224315

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

4076 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=i,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, 9131511);