Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455397 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2700 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16397 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
937 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,11,6,10)(3,13,16,24,15,8,20,23)(4,9,18,26,19,27,21,28)(14,22,17,32,30,29,31,25);(1,3)(2,8)(4,17)(5,20)(6,16)(7,15)(9,25)(10,23)(11,13)(12,24)(14,18)(19,31)(21,30)(22,28)(26,29)(27,32);(1,4)(2,9)(3,14,15,30)(5,18)(6,21)(7,19)(8,22,13,29)(10,28)(11,27)(12,26)(16,17,20,31)(23,32,24,25);(2,10,11,12)(8,23,13,24)(9,26,27,28)(22,25,29,32)Polycyclic presentation:
2^9|a^2=eg,c^2=f,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,d^c=de,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338622, #72338625, #72338633 |
| maximal normal subgroup of | #72338622, #72338625, #72338633 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368813);