Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500015 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,13)(12,20,23,16)(14,17)(15,19)(18,24,21,22);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,21)(13,17)(14,19)(16,24)(18,23)(20,22);(2,5)(3,7)(9,12)(10,16)(11,18)(13,24)(14,23)(15,22)(17,20)(19,21);(1,3,4,7)(2,6,5,8)(9,13,14,15)(10,11,17,19)(12,22,23,24)(16,21,20,18);(2,5)(6,8)(11,19)(13,15)(18,21)(22,24)Polycyclic presentation:
2^9|a^2=fi,d^2=h,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319306, #72319407 |
| maximal normal subgroup of | #72319306, #72319407 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420214);