Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(13,14,19,20,18,23,17,22)(15,24,26,28,16,21,25,27);(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(14,20)(15,16)(17,19)(21,28)(22,23)(24,27);(1,3,4,2)(6,9)(10,11)(13,15,18,16)(14,21,23,24)(17,25,19,26)(20,27,22,28);(5,7,8,12)(6,10,9,11)(13,16,18,15)(14,21,23,24)(17,26,19,25)(20,27,22,28);(15,16)(21,24)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6371304);