Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,12)(7,8,11,10)(13,14,18,21)(15,22,19,20)(16,24,26,27)(17,25,28,23);(1,2)(3,4)(5,7,9,11)(6,10,12,8)(13,15,18,19)(14,20,21,22)(16,25,26,23)(17,27,28,24);(2,4)(5,8,9,10)(6,11,12,7)(13,16,18,26)(14,17,21,28)(15,23,19,25)(20,24,22,27);(6,12)(7,11)(13,17,18,28)(14,16,21,26)(15,24,19,27)(20,23,22,25);(14,21)(16,26)(20,22)(23,25)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,e^c=ei,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, 7409212);