Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| 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 |
| prime radical | C2×#2407 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3011 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,22)(16,20)(17,26)(18,21)(19,25)(23,27)(24,28);(2,4)(5,7)(6,9)(8,11)(10,12);(6,10)(7,11)(13,15,17,25,18,24,16,23)(14,19,20,22,21,27,26,28);(7,11)(9,12)(13,16,18,17)(14,20,21,26)(15,23,24,25)(19,22,27,28);(15,24)(19,27)(22,28)(23,25)Polycyclic presentation:
2^9|c^2=gh,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72282783, #72282833, #72284158, … |
| maximal normal subgroup of | #72282783, #72282833, #72284158, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366554);