Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417530 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3047 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)(7,13)(8,17)(9,12)(10,15)(11,18)(14,19)(16,20);(5,7)(6,10)(8,14)(9,16)(11,20)(12,19)(13,17)(15,18);(10,19)(13,20);(2,4)(7,14)(8,18)(10,20)(11,17)(13,19)(15,16)Polycyclic presentation:
2^9|e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72214076, #72216013, #72216029, … |
| maximal normal subgroup of | #72214076, #72216013, #72216029, … |
| maximal quotient of | C2×#417670, #72236744, #72239552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324866);