Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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×#3060 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2053 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,9,11,12)(13,14,19,25)(15,20,26,21)(16,22,27,28)(17,24,18,23);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,20)(16,18)(17,27)(19,26)(21,25)(22,23)(24,28);(2,4)(5,7,8,11)(6,9,10,12)(13,16,19,27)(14,21,25,20)(15,17,26,18)(22,24,28,23);(6,10)(9,12)(13,15)(14,22)(16,17)(18,27)(19,26)(20,23)(21,24)(25,28);(13,17,19,18)(14,23,25,24)(15,16,26,27)(20,28,21,22)Polycyclic presentation:
2^9|a^2=fi,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,e^b=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408717);