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 | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
837 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,16,11,13,10,15)(7,17,21,27,20,30,18,12)(8,22,23,14,24,29,25,28)(19,26,34,31,33,36,32,35);(1,2,3,4)(5,7,11,20)(6,12,13,27)(8,19,24,33)(9,21,10,18)(14,26,28,36)(15,30,16,17)(22,35,29,31)(23,34,25,32);(2,4)(6,13)(7,18)(8,23)(9,10)(12,17)(14,29)(19,33)(20,21)(22,28)(24,25)(26,36)(27,30);(5,8,9,23,11,24,10,25)(6,14,16,29,13,28,15,22)(7,19,21,34,20,33,18,32)(12,26,17,31,27,36,30,35)Polycyclic presentation:
2^9|a^2=ef,b^2=fgh,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,g^c=gi,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, 380669);