Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213736 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3484 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4236 conjugacy classes of subgroups; only the 354 normal subgroups are shown.
This group has 4236 conjugacy classes of subgroups — too many to draw. Only its 354 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20);(10,16)(17,20);(2,4)(5,7,8,14,9,13,15,19)(6,10,12,17,11,16,18,20);(7,13)(8,15)(10,16)(12,18)Polycyclic presentation:
2^9|c^2=h,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^c=de,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216073, #72219299, #72219393, … |
| maximal normal subgroup of | #72216073, #72219299, #72219393, … |
| maximal quotient of | #72219310, #72225875, #72232790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289632);