Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6272 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)(11,20)(12,24)(13,18)(14,21)(15,19)(16,17)(22,23);(2,4)(5,6)(7,8)(9,11)(10,16)(12,18)(13,17)(14,23)(15,24)(19,22)(20,21);(6,8)(9,12,15,20)(10,17,19,23)(11,21,24,18)(13,16,14,22);(6,8)(9,10)(11,22)(12,17)(13,21)(14,18)(15,19)(16,24)(20,23);(9,13,15,14)(10,18,19,21)(11,23,24,17)(12,16,20,22)Polycyclic presentation:
2^9|c^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72249595, #72249596, #72249630, … |
| maximal normal subgroup of | #72249595, #72249596, #72249630, … |
| maximal quotient of | #72264445, #72305933, #72305935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344763);