Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1883 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14,20,15,19,23,18,22)(16,25,28,27,26,17,21,24);(5,6)(7,10)(8,11)(9,12)(13,15,19,22)(14,18,23,20)(16,24,26,27)(17,21,25,28);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,16,19,26)(14,17,23,25)(15,24,22,27)(18,21,20,28);(5,8,10,9)(6,11,7,12)(13,17,18,27,19,25,20,24)(14,21,22,26,23,28,15,16);(5,9,10,8)(6,11,7,12)(16,26)(17,25)(21,28)(24,27)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=g,d^2=fh,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7106386);