Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3038 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,8,7)(11,19)(12,13)(15,22)(17,18)(20,21)(23,24);(1,2,3,4)(9,10,14,16)(11,15,20,23)(12,17,13,18)(19,22,21,24);(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,22)(18,24);(5,7,8,6)(10,16)(15,23)(17,18)(22,24);(9,12,14,13)(10,17,16,18)(11,19,20,21)(15,22,23,24)Polycyclic presentation:
2^9|a^2=h,b^2=fi,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=di,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72240261, #72240346, #72261174, … |
| maximal normal subgroup of | #72240261, #72240346, #72261174, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280008);