Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 | C2×#3006 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3473 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(6,7)(9,10,14,18)(11,15,20,23)(12,16,13,17)(19,22,21,24);(5,6,8,7)(11,19)(12,13)(15,22)(16,17)(20,21)(23,24);(2,4)(5,7,8,6)(9,11)(10,15)(12,19)(13,21)(14,20)(16,22)(17,24)(18,23);(9,11)(10,15)(12,19)(13,21)(14,20)(16,22)(17,24)(18,23);(9,12,14,13)(10,16,18,17)(11,19,20,21)(15,22,23,24)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72281503, #72281929, #72282034, … |
| maximal normal subgroup of | #72281503, #72281929, #72282034, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345867);