Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,7)(10,11)(13,14,18,19)(15,16,17,20)(23,28)(24,26);(5,6,9,7)(8,11,12,10)(13,15,18,17)(14,16,19,20)(21,22,25,27)(23,26,28,24);(6,7)(10,11)(14,19)(16,20)(21,23,25,28)(22,24,27,26);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,18,20)(14,15,19,17)(21,23)(22,26)(24,27)(25,28);(5,8,9,12)(6,10,7,11)(13,17,18,15)(14,20,19,16)(21,24,25,26)(22,23,27,28)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=fh,d^2=fhi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4894203);