Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 |
| prime radical | C2×D4×#41 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6327 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)(9,10,14,19)(11,20,23,24)(12,18,13,17)(15,21,16,22);(2,4)(5,6)(7,8)(9,11)(10,15)(12,21)(13,22)(14,23)(16,19)(17,24)(18,20);(6,8)(9,11)(10,16)(12,21)(13,22)(14,23)(15,19)(17,20)(18,24);(6,8);(9,12,14,13)(10,17,19,18)(11,21,23,22)(15,24,16,20)Polycyclic presentation:
2^9|a^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=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72239231, #72249507, #72264288, … |
| maximal normal subgroup of | #72239231, #72249507, #72264288, … |
| maximal quotient of | #72249817, #72264477, #72268462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344715);