Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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×#3309 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2116 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)(13,14,19,23)(15,21,17,27)(16,25,24,20)(18,28,26,22);(5,6,8,9)(7,10,11,12)(13,15,19,17)(14,20,23,25)(16,26,24,18)(21,22,27,28);(1,2,3,4)(6,9)(7,11)(13,16)(14,20)(15,18)(17,26)(19,24)(21,22)(23,25)(27,28);(5,7)(6,10)(8,11)(9,12)(13,17,19,15)(14,21,23,27)(16,26,24,18)(20,22,25,28);(6,9)(7,11)(15,17)(16,24)(20,25)(21,27)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gi,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6513463);