Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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×#3482 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,24)(15,21,26,27)(16,20,25,28)(17,22,18,23);(1,2,3,4)(13,15,18,25,19,26,17,16)(14,20,22,27,24,28,23,21);(6,9)(10,12)(13,16)(14,20)(15,17)(18,26)(19,25)(21,22)(23,27)(24,28);(13,15,19,26)(14,21,24,27)(16,17,25,18)(20,22,28,23);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,22,24,23)(15,25,26,16)(20,21,28,27)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=i,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,e^b=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, 6361959);