Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| 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 | #35539 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2098 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)(7,10)(8,9)(11,12)(13,14,19,22)(15,20,25,28)(16,17,21,18)(23,24,26,27);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,26)(17,27)(18,24)(19,25)(21,23)(22,28);(2,4)(13,16,18,14,19,21,17,22)(15,23,24,28,25,26,27,20);(2,4)(13,15)(14,20)(16,26)(17,27)(18,24)(19,25)(21,23)(22,28);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,21,22,16)(15,24,25,27)(20,26,28,23)Polycyclic presentation:
2^9|a^2=i,c^2=g,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,d^c=di,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6352374);