Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417454 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3930 conjugacy classes of subgroups; only the 262 normal subgroups are shown.
This group has 3930 conjugacy classes of subgroups — too many to draw. Only its 262 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:
(1,2)(3,4)(5,6,7,8)(9,10,14,18)(11,15,21,24)(12,16,17,13)(19,20,23,22);(10,12)(13,16)(15,23)(17,18)(19,24)(20,22);(2,4)(9,11,14,21)(10,15,18,24)(12,19,17,23)(13,22,16,20);(6,8)(9,12)(10,16)(11,19)(13,18)(14,17)(15,20)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=hi,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72258239, #72258259, #72258262 |
| maximal normal subgroup of | #72258239, #72258259, #72258262 |
| maximal quotient of | #72267310 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356681);