Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2840 |
| commutator, Frattini, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1051 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 1051 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10)(11,17,19,14)(12,15)(13,16)(18,22,20,24)(21,23);(1,2,4,6)(3,7,8,5)(9,11)(10,14)(12,20)(13,19)(15,22)(16,17)(18,21)(23,24);(1,3,4,8)(2,5,6,7)(15,23)(22,24);(9,12)(10,15)(11,18,19,20)(13,21)(14,22,17,24)(16,23)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=h,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=def,h^b=hi,d^c=de,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72226622, #72268763, #72268778 |
| maximal normal subgroup of | #72226622, #72268763, #72268778 |
| maximal quotient of | #72210067, #72226618, #72266478, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387088);