Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,18,17)(15,19,22,25)(16,20,24,26)(21,27,28,23);(2,3)(6,9)(10,12)(13,15,18,22)(14,19,17,25)(16,23,24,27)(20,21,26,28);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,14,18,17)(15,19,22,25)(16,20,24,26)(21,27,28,23);(1,3,4,2)(13,16,18,24)(14,20,17,26)(15,21,22,28)(19,27,25,23);(6,9)(10,12)(16,24)(20,26)(21,28)(23,27)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7467425);