Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417540 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,8,11,9,12,17,20)(7,13,15,18,16,19,14,10);(5,7)(6,10)(8,14)(9,16)(11,19)(12,18)(13,20)(15,17);(10,18)(13,19);(2,4)(7,14,16,15)(10,19,18,13)Polycyclic presentation:
2^9|a^2=e,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72249969, #72250427, #72256869, … |
| maximal normal subgroup of | #72249969, #72250427, #72256869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334589);