Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3378 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)(7,12)(8,10)(9,11)(13,14,19,20)(15,17,22,18)(16,21,26,28)(23,27,24,25);(1,2)(3,4)(13,15,18,14,19,22,17,20)(16,24,25,28,26,23,27,21);(2,4)(5,7,9,8)(6,10,11,12)(14,20)(15,22)(21,28)(23,24);(2,4)(5,8,9,7)(6,10,11,12)(13,16,19,26)(14,21,20,28)(15,23,22,24)(17,27,18,25);(13,17,19,18)(14,22,20,15)(16,25,26,27)(21,23,28,24)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359297);