Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437412 |
| 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 | C2×#3036 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14)(15,23)(16,22)(17,21)(18,20)(19,24)(25,28)(26,27);(7,8)(10,12)(13,15)(14,19)(16,24)(17,26)(18,25)(20,27)(21,23)(22,28);(1,2,3,4)(5,7,9,8)(6,10,11,12)(14,20)(19,27)(21,22)(23,28);(5,8,9,7)(6,10,11,12);(13,16,18,17)(14,21,20,22)(15,24,25,26)(19,23,27,28)Polycyclic presentation:
2^9|c^2=gi,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72257377, #72264288, #72264299, … |
| maximal normal subgroup of | #72257377, #72264288, #72264299, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370051);