Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455822 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
873 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,9,18,12)(8,19)(10,15)(11,16)(13,21)(14,24,31,27)(17,20,33,23)(22,32)(25,28)(26,30)(29,36,34,35);(5,7,11,14,10,18,25,31)(6,12,28,24,15,9,16,27)(8,17,22,34,21,33,26,29)(13,20,32,36,19,23,30,35);(2,4)(5,8,10,21)(6,13,15,19)(7,17,18,33)(9,23,12,20)(11,26,25,22)(14,29,31,34)(16,32,28,30)(24,35,27,36);(5,9)(6,14)(7,16)(8,20)(10,12)(11,27)(13,29)(15,31)(17,32)(18,28)(19,34)(21,23)(22,36)(24,25)(26,35)(30,33)Polycyclic presentation:
2^9|a^2=fg,b^2=egh,c^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,d^c=de,h^c=hi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381072);