Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(9,10,11,12)(13,14)(15,23)(16,26)(17,28)(18,20)(19,25)(21,27)(22,24);(5,6)(7,8)(13,15,17,25,18,16,24,27)(14,19,22,23,20,21,28,26);(1,2,3,4)(6,8)(10,12)(14,20)(15,16)(22,28)(25,27);(13,16,17,27,18,15,24,25)(14,21,22,26,20,19,28,23)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fhi,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72308120, #72308131, #72308147, … |
| maximal normal subgroup of | #72308120, #72308131, #72308147, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281614);