Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6346594 |
| 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 | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(3,4)(7,8)(15,22)(23,24);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,24)(21,22);(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,24)(18,22);(2,7)(6,8)(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,23,22,24);(1,3,5,4)(2,6,7,8)(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,24,22,23)Polycyclic presentation:
2^9|d^2=i,e^2=hi,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,d^b=dh,f^b=fh,g^b=gi,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72249888, #72257435, #72257443, … |
| maximal normal subgroup of | #72249888, #72257435, #72257443, … |
| maximal quotient of | #72257308, #72263997, C2×#6346643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253839);