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 | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
805 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,15,11,17,10,16)(7,9,19,13,21,29,20,28)(8,14,26,32,25,34,24,33)(18,31,35,30,36,27,22,23);(1,2,3,4)(5,7)(6,13)(8,22)(9,16)(10,20)(11,21)(12,19)(14,31)(15,29)(17,28)(18,26)(23,33)(24,36)(25,35)(27,34)(30,32);(2,4)(5,8,11,25)(6,14,17,34)(7,18,21,36)(9,27,29,31)(10,26,12,24)(13,30,28,23)(15,33,16,32)(19,22,20,35);(5,9,10,28,11,29,12,13)(6,7,16,20,17,21,15,19)(8,23,24,27,25,30,26,31)(14,22,33,36,34,35,32,18)Polycyclic presentation:
2^9|a^2=ef,b^2=fgh,c^2=i,d^2=efi,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, 380672);