Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437418 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3327 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)(15,26)(16,22)(17,23)(18,24)(19,28)(20,25)(21,27);(1,2)(3,4)(13,15)(14,21)(16,19)(17,20)(18,27)(22,25)(23,24)(26,28);(2,4)(6,9)(10,12)(13,16,20,27)(14,22,25,21)(15,18,17,19)(23,28,26,24);(2,4)(13,17,20,15)(14,23,25,26)(16,18,27,19)(21,28,22,24);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,24,25,28)(15,27,17,16)(21,23,22,26)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361023);