Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,7)(9,10,16,21)(11,12,22,19)(13,20,15,18)(14,17,24,23);(1,2)(3,4)(6,7)(9,11,15,14,16,22,13,24)(10,17,20,12,21,23,18,19);(5,6,8,7)(9,12)(10,11)(13,17)(14,20)(15,23)(16,19)(18,24)(21,22);(2,4)(5,7,8,6)(9,13,16,15)(10,18,21,20)(11,14,22,24)(12,23,19,17);(9,14,13,11,16,24,15,22)(10,19,18,23,21,12,20,17)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72283039, #72283111, #72283418, … |
| maximal normal subgroup of | #72283039, #72283111, #72283418, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470430);