Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
5004 conjugacy classes of subgroups; only the 320 normal subgroups are shown.
This group has 5004 conjugacy classes of subgroups — too many to draw. Only its 320 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(9,10,11,12)(14,18)(15,16)(19,20);(5,6)(7,8)(13,14)(15,18)(16,20)(17,19);(2,4)(6,8);(10,12)(13,15)(16,17)(18,20)Polycyclic presentation:
2^9|a^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239700, #72239716, #72239750, … |
| maximal normal subgroup of | #72239700, #72239716, #72239750, … |
| maximal quotient of | #72249893, #72257159, #72321996 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356660);