#9839208

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

4125 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9746404);