Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437415 |
| 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 |
| prime radical | #23924 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3267 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,20)(15,19,22,26)(16,18,24,21)(23,25,28,27);(1,2)(3,4)(14,18)(16,24)(19,25)(20,21)(23,28)(26,27);(2,4)(6,9)(10,12)(13,15,17,22)(14,19,20,26)(16,23,24,28)(18,25,21,27);(2,4)(15,22)(19,26)(23,28)(25,27);(5,7,8,11)(6,10,9,12)(14,20)(18,21)(19,26)(25,27)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72281938, #72282147 |
| maximal normal subgroup of | #72281938, #72282147 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360944);