Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573549 |
| 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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2831 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)(7,12)(8,10)(9,11)(13,14,18,22)(15,19,23,27)(16,21,17,20)(24,26,25,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,24)(17,25)(18,23)(20,28)(21,26)(22,27);(2,4)(14,20)(15,23)(16,17)(19,26)(21,22)(27,28);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(13,16,18,17)(14,21,22,20)(15,24,23,25)(19,26,27,28);(2,4)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,e^d=ehi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483894);