Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
2524 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)(7,12)(8,10)(9,11)(13,14,18,16)(15,22,23,28)(17,21,19,27)(20,25,24,26);(2,4)(7,8)(10,12)(14,16)(15,23)(21,27)(25,26);(5,7,9,8)(6,10,11,12)(13,15,17,25,18,23,19,26)(14,20,27,28,16,24,21,22);(5,8,9,7)(6,10,11,12)(13,16,18,14)(15,24,23,20)(17,21,19,27)(22,26,28,25);(13,17,18,19)(14,21,16,27)(15,25,23,26)(20,22,24,28)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370021);