Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C2×C42 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | #1820 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
672 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 672 conjugacy classes of subgroups — too many to draw. Only its 150 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 — 3 generators on 36 points:
(5,6)(7,17,10,15)(8,20)(9,14)(11,16)(12,19,28,23)(13,18,30,25)(21,33)(22,31)(24,27)(26,34)(29,32,36,35);(1,2,3,4)(5,7,11,10)(6,12)(8,18,22,25)(9,19,24,23)(13,26)(14,17)(15,27)(16,28)(20,29)(21,32,34,35)(30,33)(31,36);(5,8,9,21,11,22,24,34)(6,13,14,29,16,30,27,36)(7,18,19,32,10,25,23,35)(12,26,17,20,28,33,15,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gi,h^a=hi,f^b=fi,e^c=ei,f^c=fhi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281189, #72314009, #72314013 |
| maximal normal subgroup of | #72281189, #72314009, #72314013 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54066);