Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3280 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)(14,20)(15,22)(17,18)(21,27)(23,28)(25,26);(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,21)(18,20)(19,22)(23,25)(24,28)(26,27);(6,9)(7,11)(13,15,19,20)(14,17,22,18)(16,23,24,27)(21,25,28,26);(5,7,8,11)(6,10,9,12)(13,16,19,24)(14,21,22,28)(15,23,20,27)(17,25,18,26);(6,9)(10,12)(16,24)(21,28)(23,27)(25,26)Polycyclic presentation:
2^9|c^2=i,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420061);