Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417742 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15257 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,13)(4,18)(5,6)(10,26)(11,12)(14,24)(16,17)(20,21)(23,31)(27,29)(28,32);(1,2)(3,14)(4,19)(5,11)(6,8)(7,12)(9,16)(10,18)(13,25)(15,23)(17,24)(20,22)(21,26)(27,28)(29,31)(30,32);(1,3)(2,9)(4,15)(5,16)(6,13)(7,17)(8,14)(10,23)(11,24)(12,25)(18,27)(19,31)(20,30)(21,29)(22,28)(26,32);(1,4)(2,10)(3,15)(5,21)(6,18)(7,20)(8,22)(9,23)(11,19)(12,26)(13,27)(14,28)(16,29)(17,30)(24,31)(25,32)Polycyclic presentation:
2^9|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,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72338545, #72338553, #72338560 |
| maximal normal subgroup of | #72338545, #72338553, #72338560 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324938);