Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431707 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
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 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
7350 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)(9,10)(11,19)(12,21)(13,16)(14,17)(15,18)(20,23)(22,24);(2,4)(5,6)(7,8)(9,11)(10,12)(13,21)(14,20)(15,22)(16,24)(17,19)(18,23);(6,8)(9,12)(10,11)(13,19)(14,24)(15,23)(16,20)(17,21)(18,22);(9,13,15,14)(10,16,18,17)(11,20,22,21)(12,19,23,24);(9,13,15,14)(10,17,18,16)(11,21,22,20)(12,19,23,24)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249555, #72249572, #72249596, … |
| maximal normal subgroup of | #72249555, #72249572, #72249596, … |
| maximal quotient of | #72305936, #72311158, #72316005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338964);