Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #156684 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C24 |
| prime radical | C26 | C23 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
3696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,7,8)(10,12)(13,14)(15,17)(16,19)(18,20)(21,22,24,25)(23,26,27,28);(2,4)(5,6,7,8)(14,15)(16,20)(22,25)(26,28);(5,6)(7,8)(9,10)(11,12)(13,15,17,14)(16,19,20,18)(21,23,24,27)(22,26,25,28);(1,2)(3,4)(5,6)(7,8)(13,16)(14,18)(15,19)(17,20)(21,22)(23,26)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=eg,b^2=eh,c^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72328527, #72328538, #72328544, … |
| maximal normal subgroup of | #72328527, #72328538, #72328544, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 64473);