Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3352 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)(6,8)(9,10,16,20,15,19,14,13)(11,21,12,17,23,24,22,18);(2,4)(9,11)(10,17)(12,14)(13,24)(15,23)(16,22)(18,19)(20,21);(5,6,7,8)(9,12,14,11,15,22,16,23)(10,18,13,24,19,17,20,21);(9,13,16,10,15,20,14,19)(11,17,12,24,23,18,22,21);(9,14,15,16)(10,13,19,20)(11,12,23,22)(17,24,18,21)Polycyclic presentation:
2^9|a^2=fi,c^2=gh,d^2=h,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72238790, #72239652, #72239910, … |
| maximal normal subgroup of | #72238790, #72239652, #72239910, … |
| maximal quotient of | #72238785 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372852);