Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 | |
3029 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)(7,11)(10,12)(13,14,19,23,20,24,18,22)(15,17,16,21,26,28,25,27);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,18)(17,23)(19,25)(20,26)(22,28)(24,27);(6,9)(10,12)(13,16,20,25)(14,17,24,28)(15,18,26,19)(21,23,27,22);(6,9)(10,12)(13,17,20,28)(14,16,24,25)(15,23,26,22)(18,27,19,21);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,22,24,23)(15,16,26,25)(17,21,28,27)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398717);