Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418818 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15467 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1547 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12,14,7)(4,16,18,8)(5,11)(6,10)(9,15,26,20)(13,22)(17,27)(19,30,32,24)(21,25)(23,28)(29,31);(1,3)(2,7)(4,17,18,13)(5,20)(6,14)(8,22,16,27)(9,25)(10,12)(11,26)(15,21)(19,28,32,31)(23,24,29,30);(1,4)(2,8)(3,13)(5,19)(6,18)(7,22)(9,23)(10,16)(11,24)(12,27)(14,17)(15,28)(20,31)(21,32)(25,30)(26,29);(1,3)(2,9)(4,13)(5,20)(6,14)(7,11)(8,23)(10,26)(12,25)(15,21)(16,29)(17,18)(19,31)(22,24)(27,30)(28,32)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285635, #72338513, #72338518, … |
| maximal normal subgroup of | #72285635, #72338513, #72338518, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340042);