Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| 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 | |
2047 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)(5,6)(7,11)(8,10)(9,12)(13,14,19,23)(15,25,16,20)(17,24,27,21)(18,22,26,28);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,19)(17,26)(18,27)(21,28)(22,24)(23,25);(1,2,3,4)(13,16,27,18,19,15,17,26)(14,20,21,22,23,25,24,28);(6,10)(9,11)(13,17,19,27)(14,21,23,24)(15,26,16,18)(20,28,25,22);(7,12)(9,11)(13,18,17,16,19,26,27,15)(14,22,24,20,23,28,21,25)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72308353 |
| maximal normal subgroup of | #72308353 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6721632);