Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3230 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)(13,14,16,25,18,21,17,20)(15,22,26,19,23,28,24,27);(2,4)(5,6,8,9)(7,10,11,12)(13,15,18,23)(14,19,21,27)(16,24,17,26)(20,28,25,22);(6,9)(7,11)(13,14,18,21)(15,19,23,27)(16,20,17,25)(22,24,28,26);(5,7)(6,10)(8,11)(9,12)(15,23)(19,27)(22,28)(24,26);(6,9)(7,11)(13,16,18,17)(14,20,21,25)(15,24,23,26)(19,28,27,22)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503733);