Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1110 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 1110 conjugacy classes of subgroups — too many to draw. Only its 242 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 — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,9)(7,12,8,11)(13,14,16,18)(15,17,19,20)(21,22,26,30)(23,28,32,36)(24,33,31,27)(25,34,35,29);(5,7,10,8)(6,11,9,12)(13,15)(14,17)(16,19)(18,20)(21,23,25,31,26,32,35,24)(22,27,29,28,30,33,34,36);(2,4)(5,8,10,7)(6,11,9,12)(15,19)(17,20)(23,24)(25,35)(27,36)(28,33)(29,34)(31,32);(5,9,10,6)(7,12,8,11)(21,24,25,23,26,31,35,32)(22,28,29,33,30,36,34,27)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=ehi,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 382736);