Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 | |
4826 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)(9,10,14,18)(11,15,21,23)(12,16,17,13)(19,24,22,20);(2,4)(10,12)(11,19)(13,16)(15,23)(17,18)(21,22);(5,6,7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,24)(18,23);(9,12)(10,16)(11,20)(13,18)(14,17)(15,19)(21,24)(22,23);(11,21)(15,23)(19,22)(20,24)Polycyclic presentation:
2^9|a^2=i,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72249595, #72249659, #72305339, … |
| maximal normal subgroup of | #72249595, #72249659, #72305339, … |
| maximal quotient of | #72264485, #72305985, #72305986, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349873);