#9923650

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

4177 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)(3,10)(4,9)(5,7)(6,8)(11,16)(12,15)(13,14)(17,18,22,23)(19,21,25,30)(20,24,28,31)(26,29,32,27)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(1,3,6,5)(2,7,8,10)(4,11,12,13)(9,14,15,16)(17,19)(18,21)(20,26)(22,25)(23,30)(24,29)(27,31)(28,32)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(2,8)(7,10)(9,15)(14,16)(18,23)(19,25)(24,31)(26,32)(33,35,39,36)(34,41,37,40)(38,43,46,44)(42,48,45,47);(1,2)(3,7)(4,9)(5,10)(6,8)(11,14)(12,15)(13,16)(17,18)(19,21)(20,24)(22,23)(25,30)(26,29)(27,32)(28,31)(33,36,39,35)(34,40,37,41)(38,44,46,43)(42,47,45,48);(1,4)(2,9)(3,11)(5,13)(6,12)(7,14)(8,15)(10,16)(17,20,22,28)(18,24,23,31)(19,26,25,32)(21,29,30,27)(33,37,39,34)(35,40,36,41)(38,42,46,45)(43,48,44,47);(1,5,6,3)(2,10,8,7)(4,11,12,13)(9,14,15,16)(17,21)(18,19)(20,27)(22,30)(23,25)(24,32)(26,31)(28,29)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=ghi,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, 9830846);