Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469995 |
| 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×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,25,17,22)(16,23,18,21)(20,26,28,27);(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,27)(17,19)(18,26)(21,25)(22,23)(24,28);(1,2,3,4)(7,11)(9,12)(13,16,19,18)(14,21,24,23)(15,26,17,27)(20,22,28,25);(7,11)(9,12)(13,17,19,15)(14,22,24,25)(16,27,18,26)(20,21,28,23);(6,10)(9,12)(13,16,19,18)(14,21,24,23)(15,27,17,26)(20,25,28,22)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6380315);