Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993979 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,17,20,18,21,16,25)(15,22,24,27,23,28,26,19);(2,3)(5,6)(7,11)(8,10)(9,12)(15,23)(19,27)(22,28)(24,26);(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,24)(17,26)(18,23)(20,28)(21,27)(22,25);(1,3,4,2)(7,12)(9,11)(15,23)(19,27)(22,28)(24,26);(6,10)(9,11)(13,16,18,17)(14,20,21,25)(15,24,23,26)(19,28,27,22)Polycyclic presentation:
2^9|a^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6913428);