Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1863 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)(7,8)(9,11)(10,12)(13,14,19,21,18,20,27,28)(15,16,23,24,22,17,25,26);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,17)(16,20)(18,22)(19,23)(21,26)(24,28)(25,27);(2,4)(5,8,9,10)(6,7,11,12)(13,16,18,17)(14,15,20,22)(19,26,27,24)(21,25,28,23);(6,11)(7,12)(13,17,18,16)(14,15,20,22)(19,24,27,26)(21,25,28,23);(14,20)(15,22)(21,28)(23,25)Polycyclic presentation:
2^9|a^2=g,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,e^b=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, 7410617);