Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| 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 | #23428 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2313 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,12)(7,8,11,10)(13,14,17,22,18,21,15,19)(16,24,23,27,25,28,26,20);(1,2)(3,4)(5,7,9,11)(6,10,12,8)(13,15,18,17)(14,19,21,22)(16,23,25,26)(20,24,27,28);(2,4)(5,8,9,10)(6,11,12,7)(14,19)(15,17)(20,24)(21,22)(23,26)(27,28);(6,12)(7,11)(13,16,18,25)(14,20,21,27)(15,23,17,26)(19,24,22,28);(14,21)(19,22)(20,27)(24,28)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410996);