Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #97030 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1150 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1150 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) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 44 points:
(5,6)(7,11,12,9)(8,10)(13,14)(15,24,28,37)(16,29)(17,22)(18,33)(19,23)(20,26,35,27)(21,31)(25,34,42,44)(30,36)(32,38)(39,40,43,41);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,25)(17,27)(18,26)(19,28)(21,34)(22,37)(23,35)(24,33)(29,39)(30,41)(31,40)(32,42)(36,44)(38,43);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,32)(14,21,23,36)(15,25,28,42)(17,31,18,30)(20,34,35,44)(22,38,33,29)(24,39,37,43)(26,41,27,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,d^c=di,e^c=ei,g^c=gh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18594);