Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32870 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2826 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14622 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
920 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,14)(3,15,17,25)(4,18)(5,11,9,26)(6,13,7,12)(10,21)(16,24)(19,20)(22,31)(23,30)(27,29)(28,32);(1,3,6,11,8,17,7,26)(2,9,12,15,14,5,13,25)(4,19,21,29,23,32,22,16)(10,28,18,27,31,20,30,24);(1,4)(2,10)(3,16)(5,24)(6,22)(7,21)(8,23)(9,27)(11,32)(12,30)(13,18)(14,31)(15,28)(17,29)(19,26)(20,25);(1,5)(2,11)(3,12)(4,20)(6,15)(7,25)(8,9)(10,29)(13,17)(14,26)(16,31)(18,19)(21,27)(22,24)(23,28)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,d^b=def,f^b=fi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72319162, #72319352, #72319366, … |
| maximal normal subgroup of | #72319162, #72319352, #72319366, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298445);