Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,10,11,12)(13,14,16,21,18,23,17,22)(15,24,26,20,27,28,25,19);(1,2)(3,4)(6,9)(10,12)(13,15)(14,19)(16,25)(17,26)(18,27)(20,23)(21,28)(22,24);(2,4)(13,15)(14,20)(16,25)(17,26)(18,27)(19,23)(21,24)(22,28);(5,7,8,11)(6,10,9,12);(13,16,18,17)(14,21,23,22)(15,25,27,26)(19,28,20,24)Polycyclic presentation:
2^9|a^2=f,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72264217, #72264363, #72264439, … |
| maximal normal subgroup of | #72264217, #72264363, #72264439, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339786);