#9913669

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

4182 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)(3,11)(4,10)(5,8)(6,14)(7,9)(12,13)(15,16)(17,18,22,25)(19,21,27,23)(20,24,28,31)(26,29,32,30)(33,34,38,42)(35,39,44,48)(36,45,46,40)(37,41,43,47);(1,3,7,5)(2,8,9,11)(4,12,14,15)(6,13,10,16)(20,28)(24,31)(26,32)(29,30)(33,35,38,44)(34,39,42,48)(36,43,46,37)(40,41,45,47);(2,9)(4,14)(8,11)(12,15)(17,19,22,27)(18,23,25,21)(20,26,28,32)(24,30,31,29)(33,36,38,46)(34,40,42,45)(35,37,44,43)(39,47,48,41);(1,4)(2,10)(3,12)(5,15)(6,9)(7,14)(8,16)(11,13)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,32)(30,31)(33,36)(34,40)(35,43)(37,44)(38,46)(39,41)(42,45)(47,48);(1,5,7,3)(2,8,9,11)(4,15,14,12)(6,13,10,16)(17,20,22,28)(18,24,25,31)(19,26,27,32)(21,29,23,30)(33,37,38,43)(34,41,42,47)(35,36,44,46)(39,45,48,40);(1,6,7,10)(2,4,9,14)(3,13,5,16)(8,12,11,15)(17,21)(18,19)(20,29)(22,23)(24,26)(25,27)(28,30)(31,32)(33,34)(35,39)(36,40)(37,47)(38,42)(41,43)(44,48)(45,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,e^2=h,f^2=g,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, 9820865);