#9923909

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

4005 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9831105);