Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2934 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)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,19)(15,23,25,20)(16,17,22,21)(24,27,28,26);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,24)(17,26)(18,25)(19,23)(21,27)(22,28);(2,4)(14,19)(15,23)(17,21)(20,25)(24,26)(27,28);(6,10)(9,11)(13,16,18,22)(14,21,19,17)(15,24,25,28)(20,27,23,26);(7,12)(9,11)(13,17,18,21)(14,22,19,16)(15,24,25,28)(20,26,23,27)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6574697);