#9726229

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

4092 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)(4,9)(5,13)(6,7)(10,15)(11,12)(14,16)(17,18)(19,21)(20,27)(22,30)(23,24)(25,32)(26,31)(28,29)(33,34)(35,44)(36,46)(37,41)(38,45)(39,40)(42,47)(43,48);(2,7)(4,8)(10,13)(12,14)(17,19,23,20)(18,21,24,27)(22,26,32,28)(25,29,30,31)(34,40)(35,37)(42,46)(43,45);(2,7)(3,9)(10,13)(11,16)(18,24)(19,20)(25,30)(26,28)(33,35,39,37)(34,41,40,44)(36,43,47,45)(38,42,48,46);(1,3,6,9)(2,8,7,4)(5,11,15,16)(10,14,13,12)(17,20,23,19)(18,21,24,27)(22,28,32,26)(25,29,30,31)(33,36,39,47)(34,42,40,46)(35,45,37,43)(38,44,48,41);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,13,16)(17,21)(18,19)(20,24)(22,31)(23,27)(25,28)(26,30)(29,32)(33,37,39,35)(34,41,40,44)(36,43,47,45)(38,46,48,42);(1,5,6,15)(2,10,7,13)(3,11,9,16)(4,12,8,14)(17,22,23,32)(18,25,24,30)(19,26,20,28)(21,29,27,31)(33,38)(34,43)(35,42)(36,44)(37,46)(39,48)(40,45)(41,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9633425);