Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5039 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)(10,15)(12,13)(16,22)(17,21)(18,20)(23,24);(5,6)(7,8)(9,10,14,17)(11,16,19,24)(12,15,13,21)(18,23,20,22);(2,4)(6,8)(9,11)(10,16)(12,20)(13,18)(14,19)(15,22)(17,24)(21,23);(5,6)(7,8)(9,12,14,13)(10,15,17,21)(11,18,19,20)(16,23,24,22);(6,8)(11,19)(16,24)(18,20)(22,23)Polycyclic presentation:
2^9|b^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72249712, #72256835, #72256955, … |
| maximal normal subgroup of | #72249712, #72256835, #72256955, … |
| maximal quotient of | #72249771, #72305495, #72315911 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480816);