Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994043 |
| 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 | |
2146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,17,22,18,21,15,19)(16,24,23,27,25,28,26,20);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,17)(14,19,21,22)(16,23,25,26)(20,24,27,28);(5,7)(6,9)(8,12)(10,11)(13,16)(14,20)(15,23)(17,26)(18,25)(19,24)(21,27)(22,28);(1,3,4,2)(7,12)(9,11)(14,21)(19,22)(20,27)(24,28);(6,10)(9,11)(13,15,18,17)(14,19,21,22)(16,23,25,26)(20,24,27,28)Polycyclic presentation:
2^9|a^2=fg,b^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,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, 6913628);