Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3733 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,17)(16,18)(19,20);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11);(2,4)(14,17)(15,20)(18,19);(6,10)(7,12)(14,17)(15,20)(18,19);(14,18)(17,19)Polycyclic presentation:
2^9|g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72237210, #72239595, #72249607, … |
| maximal normal subgroup of | #72237210, #72239595, #72249607, … |
| maximal quotient of | #72238935, #72239738, #72239747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407857);