Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#546 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,14)(15,25)(16,22)(17,24)(18,23)(19,21)(20,26)(27,28);(5,7)(6,9)(8,12)(10,11)(13,15,18,26)(14,20,23,25)(16,19,27,17)(21,22,24,28);(14,21)(15,26)(17,19)(20,28)(22,25)(23,24);(7,12)(9,11)(13,16,18,27)(14,22,23,28)(15,17,26,19)(20,21,25,24);(1,3,4,2)(6,10)(9,11)(13,17,18,19)(14,21,23,24)(15,27,26,16)(20,28,25,22)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^b=ei,e^c=ei,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6306129);