Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511802 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #3497 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1050 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 152 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:
(2,4)(5,6)(7,13)(8,17)(9,25)(10,15)(11,27)(12,18)(14,24)(16,26)(19,34)(20,28)(21,29)(22,32)(23,31)(30,35)(33,36);(1,2,3,4)(5,7)(6,13)(8,23)(9,20)(10,21)(11,19)(12,22)(14,33)(15,29)(16,30)(17,31)(18,32)(24,36)(25,28)(26,35)(27,34);(5,8,12,11)(6,14,18,25)(7,19,22,23)(9,26,24,10)(13,28,32,33)(15,17,16,27)(20,21,36,35)(29,34,30,31);(5,9,12,24)(6,15,18,16)(7,20,22,36)(8,10,11,26)(13,29,32,30)(14,27,25,17)(19,35,23,21)(28,31,33,34)Polycyclic presentation:
2^9|b^2=e,c^2=i,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,h^a=hi,c^b=ci,d^c=di,f^c=fi,g^c=ghi,f^d=fh,g^d=gi,g^f=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419125);