#9160780

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	#50278	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4096 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,d^2=hi,e^2=gh,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^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, 9067976);