Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3493 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,18,22)(15,17,19,20)(16,21,25,28)(23,26,27,24);(2,4)(5,7,9,8)(6,10,12,11)(14,19)(15,22)(16,24)(17,20)(21,28)(25,26);(5,7,9,8)(6,11,12,10)(13,15,18,19)(14,20,22,17)(16,23,25,27)(21,24,28,26);(5,8,9,7)(6,11,12,10)(13,16,18,25)(14,21,22,28)(15,23,19,27)(17,26,20,24);(14,22)(15,19)(21,28)(23,27)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=g,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360224);