Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395434 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14724 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3585 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3453 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)(3,13)(4,10,19,26)(5,11)(6,9)(7,12)(8,14)(15,29,31,27)(16,21)(17,23)(18,24,20,25)(22,28,32,30);(1,3)(2,8)(4,15)(5,16)(6,14)(7,17)(9,23)(10,22)(11,13)(12,21)(18,28)(19,31)(20,30)(24,29)(25,27)(26,32);(2,9)(3,14)(5,6)(8,21)(10,24)(11,12)(15,28)(16,17)(18,20)(22,32)(25,26)(30,31);(1,4)(2,10)(3,15)(5,20)(6,18)(7,19)(8,22)(9,24)(11,25)(12,26)(13,27)(14,28)(16,30)(17,31)(21,32)(23,29)Polycyclic presentation:
2^9|a^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215906, #72219682, #72219689, … |
| maximal normal subgroup of | #72215906, #72219682, #72219689, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302630);