Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(14,18)(15,21)(16,24)(19,23)(22,26)(27,28);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,14,17,18)(15,21,23,19)(16,20,24,25)(22,27,28,26);(2,4)(13,15,14,19,17,23,18,21)(16,22,20,26,24,28,25,27);(6,10)(9,11)(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28);(2,4)(7,12)(9,11)(16,24)(20,25)(22,28)(26,27)Polycyclic presentation:
2^9|b^2=i,c^2=gi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589560);