Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470922 |
| 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×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1955 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(7,11)(10,12)(13,14,19,22)(15,20,24,26)(16,25,27,23)(17,21,18,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,21)(17,23)(18,25)(19,24)(22,26)(27,28);(6,9)(10,12)(13,16,18,26,19,27,17,20)(14,15,21,25,22,24,28,23);(6,9)(10,12)(13,17,19,18)(14,21,22,28)(15,23,24,25)(16,26,27,20);(5,7,8,11)(6,10,9,12)(15,24)(16,27)(20,26)(23,25)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6401325);