#9923640

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

4183 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)(3,10)(4,9)(5,14)(6,8)(7,12)(11,16)(13,15)(17,18,22,23)(19,21,25,30)(20,24,28,31)(26,29,32,27)(34,38)(36,44)(39,41)(42,48);(1,3)(2,7)(4,11)(5,9)(6,12)(8,10)(13,14)(15,16)(17,19)(18,21)(20,26)(22,25)(23,30)(24,29)(27,31)(28,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,8)(4,13)(7,10)(11,14)(18,23)(19,25)(24,31)(26,32)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,4,6,13)(2,9,8,15)(3,11,12,14)(5,10,16,7)(17,18)(19,21)(20,24)(22,23)(25,30)(26,29)(27,32)(28,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,3)(2,10)(4,11)(5,15)(6,12)(7,8)(9,16)(13,14)(17,20,22,28)(18,24,23,31)(19,26,25,32)(21,29,30,27)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5,6,16)(2,11,8,14)(3,9,12,15)(4,10,13,7)(17,21)(18,19)(20,27)(22,30)(23,25)(24,32)(26,31)(28,29)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=gh,c^2=ghi,d^2=hi,e^2=gh,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, 9830836);