#9221169

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
prime radical	#49589	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4209 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)(3,8)(4,6)(5,9)(7,11)(10,16)(12,14)(13,15)(17,18,23,25)(19,20,27,21)(22,24,31,32)(26,28,30,29)(33,34,38,42)(35,43,46,39)(36,40,45,47)(37,48,44,41);(1,3)(2,7)(4,8)(5,10)(6,11)(9,15)(12,16)(13,14)(17,19)(18,20)(21,25)(22,30)(23,27)(24,29)(26,31)(28,32)(33,35,38,46)(34,39,42,43)(36,37,45,44)(40,41,47,48);(17,20,23,21)(18,19,25,27)(22,28,31,29)(24,26,32,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,48)(45,47);(1,4,6,2)(3,8,11,7)(5,9,14,12)(10,15,13,16)(17,21,23,20)(18,19,25,27)(22,28,31,29)(24,30,32,26)(33,34)(35,39)(36,47)(37,48)(38,42)(40,45)(41,44)(43,46);(1,3)(2,8)(4,7)(5,13)(6,11)(9,15)(10,14)(12,16)(17,22)(18,24)(19,26)(20,28)(21,29)(23,31)(25,32)(27,30)(33,36)(34,40)(35,44)(37,46)(38,45)(39,48)(41,43)(42,47);(1,5,6,14)(2,9,4,12)(3,10,11,13)(7,15,8,16)(17,19)(18,21)(20,25)(22,30)(23,27)(24,28)(26,31)(29,32)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,47,43,40)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,d^2=gh,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, 9128365);