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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1952 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)(7,8)(9,11)(10,12)(13,14,17,20,16,18,24,26)(15,21,23,27,22,28,25,19);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(14,18)(19,27)(20,26)(21,28);(2,4)(5,8,9,10)(6,7,11,12)(13,15,16,22)(14,19,18,27)(17,25,24,23)(20,28,26,21);(6,11)(7,12)(13,14,16,18)(15,19,22,27)(17,26,24,20)(21,23,28,25);(15,22)(19,27)(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,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, 7410681);