Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419843 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| 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 | |
1017 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)(5,6,10,14,9,15,11,16)(7,17,20,28,19,12,21,32)(8,22,24,30,25,13,26,31)(18,27,23,36,34,33,35,29);(5,7)(6,12)(8,23,25,35)(9,19)(10,20)(11,21)(13,29,22,36)(14,32)(15,17)(16,28)(18,24,34,26)(27,30,33,31);(5,8)(6,13,15,22)(7,18)(9,25)(10,26)(11,24)(12,27,17,33)(14,30,16,31)(19,34)(20,35)(21,23)(28,36,32,29);(2,4)(6,14)(7,19)(10,11)(12,28)(13,30)(15,16)(17,32)(18,34)(22,31)(24,26)(27,36)(29,33)Polycyclic presentation:
2^9|a^2=efg,b^2=ei,c^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285470, #72285492, #72285577, … |
| maximal normal subgroup of | #72285470, #72285492, #72285577, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342030);