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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1924 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,11)(7,8,12,10)(13,14,18,20)(15,22,25,28)(16,21,26,27)(17,24,19,23);(1,2)(3,4)(5,7,9,12)(6,10,11,8)(14,20)(15,25)(16,26)(23,24);(2,4)(13,15)(14,21)(16,17)(18,25)(19,26)(20,27)(22,23)(24,28);(5,8,9,10)(6,7,11,12)(13,16,18,26)(14,22,20,28)(15,17,25,19)(21,23,27,24);(13,17,18,19)(14,23,20,24)(15,16,25,26)(21,22,27,28)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=f,e^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,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, 7409645);