Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #45319 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2085 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)(5,6)(7,11)(8,9)(10,12)(13,14,18,21)(15,22,25,28)(16,19,27,26)(17,24,23,20);(2,3)(5,6)(7,10)(8,9)(11,12)(13,15,23,19,18,25,17,26)(14,16,24,28,21,27,20,22);(1,2,4,3)(6,9)(10,11)(13,16,18,27)(14,19,21,26)(15,24,25,20)(17,22,23,28);(1,3,4,2)(5,7,8,12)(6,10,9,11)(15,25)(16,27)(19,26)(22,28);(13,15,23,19,18,25,17,26)(14,16,24,28,21,27,20,22)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=h,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504172);