Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2152 conjugacy classes of subgroups; only the 626 normal subgroups are shown.
This group has 2152 conjugacy classes of subgroups — too many to draw. Only its 626 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) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22,26,30)(23,31,32,36)(24,29,25,28)(27,34,35,33);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23,24,34,26,32,25,33)(22,27,28,36,30,35,29,31);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(23,32)(27,35)(31,36)(33,34);(6,11)(10,16)(12,18)(17,20)(21,24,26,25)(22,28,30,29)(23,33,32,34)(27,31,35,36);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(23,32)(27,35)(31,36)(33,34)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,d^b=di,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344426);