Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2428 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14207 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1406 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)(4,18,21,27)(5,10)(6,11)(7,12)(8,15)(9,20,28,22)(14,23,19,26)(16,25)(17,24)(29,30,32,31);(1,3)(2,8,12,25)(4,19,21,14)(5,15)(6,16)(7,17)(9,26)(10,24,11,13)(18,29)(20,31,22,30)(23,28)(27,32);(1,4)(2,9)(3,14)(5,20)(6,22)(7,21)(8,23)(10,18)(11,27)(12,28)(13,29)(15,30)(16,31)(17,19)(24,32)(25,26);(2,10,12,11)(8,24,25,13)(9,18,28,27)(23,32,26,29)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=efh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213889, #72219488, #72219491, … |
| maximal normal subgroup of | #72213889, #72219488, #72219491, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285372);