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 | C2×#3179 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1827 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,7)(8,11,12,10)(13,14,17,19)(15,21,24,28)(16,26,22,20)(18,27,23,25);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,15,22,27,17,24,16,25)(14,18,26,28,19,23,20,21);(5,8,9,12)(6,10,7,11)(13,14)(15,23)(16,20)(17,19)(18,24)(21,27)(22,26)(25,28);(2,4)(6,7)(10,11)(14,19)(15,24)(20,26)(25,27);(13,15,22,27,17,24,16,25)(14,18,26,28,19,23,20,21)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=gi,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^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7453113);