Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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 | #38909 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(13,14,18,20)(15,22,23,25)(16,24,19,27)(17,26,28,21);(2,4)(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,20)(17,27)(18,23)(21,25)(22,26)(24,28);(6,9)(7,11)(13,16,18,19)(14,15,20,23)(17,22,28,25)(21,27,26,24);(5,7)(6,10)(8,11)(9,12)(14,20)(15,23)(21,26)(24,27);(6,9)(7,11)(14,20)(15,23)(21,26)(24,27)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6502958);