Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454681 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| 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 | |
958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17)(8,22,26,28)(9,12)(10,15)(11,16)(13,18)(14,24,32,25)(19,30,34,35)(20,29)(21,27)(23,36,33,31);(1,2,3,4)(6,12)(7,18)(9,10)(13,29)(14,28)(15,16)(19,23)(20,21)(22,32)(24,25)(30,35)(33,34);(2,4)(5,7,11,21)(6,13,16,29)(8,23,26,33)(9,18,10,20)(12,27,15,17)(14,31,32,36)(19,25,34,24)(22,35,28,30);(5,8)(6,14)(7,19)(9,25)(10,24)(11,26)(12,28)(13,30)(15,22)(16,32)(17,31)(18,23)(20,33)(21,34)(27,36)(29,35)Polycyclic presentation:
2^9|a^2=ef,b^2=gh,c^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,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, 378984);