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×#3486 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3440 conjugacy classes of subgroups; only the 354 normal subgroups are shown.
This group has 3440 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) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,11)(8,12)(9,14)(10,13)(15,24)(16,23)(17,25)(18,26)(19,28)(20,27)(21,30)(22,29)(31,35)(32,34)(33,36);(11,23)(24,26)(25,34)(35,36);(2,4)(5,7,9,17,10,16,22,32)(6,11,14,25,13,23,29,34)(8,18,19,33,20,15,21,31)(12,26,28,36,27,24,30,35);(5,8,10,20)(6,12,13,27)(7,15,16,18)(9,21,22,19)(11,24,23,26)(14,30,29,28)(17,33,32,31)(25,36,34,35)Polycyclic presentation:
2^9|c^2=h,d^2=i,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 | #72306354, #72306508, #72306519 |
| maximal normal subgroup of | #72306354, #72306508, #72306519 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289634);