#9924016

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

4027 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)(3,9)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,28)(20,30)(21,25)(22,29)(23,24)(26,31)(27,32)(33,34,38,39)(35,36,40,43)(37,41,46,47)(42,44,48,45);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,13)(15,16)(18,24)(19,21)(26,30)(27,29)(34,39)(36,43)(41,47)(44,45);(2,7)(6,9)(8,15)(14,16)(17,19,23,21)(18,25,24,28)(20,27,31,29)(22,26,32,30)(33,35,38,40)(34,36,39,43)(37,42,46,48)(41,44,47,45);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,20,23,31)(18,26,24,30)(19,29,21,27)(22,28,32,25);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,21,23,19)(18,25,24,28)(20,27,31,29)(22,30,32,26)(33,36)(34,40)(35,39)(37,45)(38,43)(41,42)(44,46)(47,48);(1,3)(2,6)(4,12)(5,11)(7,9)(8,16)(10,13)(14,15)(17,22)(18,27)(19,26)(20,28)(21,30)(23,32)(24,29)(25,31)(33,37,38,46)(34,41,39,47)(35,42,40,48)(36,44,43,45)

Polycyclic presentation:

2^9|a^2=gh,c^2=ghi,d^2=i,e^2=h,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, 9831212);