Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417626 |
| 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 | |
2375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,13)(4,18)(5,6)(8,14)(9,23)(10,11)(16,17)(19,20)(22,26)(24,25)(27,31)(28,29);(1,2)(3,14)(4,9)(5,10)(6,11)(7,12)(8,17)(13,22)(15,29)(16,26)(18,23)(19,25)(20,24)(21,27)(28,30)(31,32);(1,3)(2,8)(4,15)(5,17)(6,13)(7,16)(9,21)(10,26)(11,14)(12,22)(18,31)(19,30)(20,27)(23,29)(24,28)(25,32);(1,4)(2,9)(3,15)(5,18)(6,20)(7,19)(8,21)(10,23)(11,24)(12,25)(13,27)(14,28)(16,30)(17,31)(22,32)(26,29)Polycyclic presentation:
2^9|f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215958, #72216013, #72216071, … |
| maximal normal subgroup of | #72215958, #72216013, #72216071, … |
| maximal quotient of | C2×#417626, #72252191, #72313941 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324822);