Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(10,14)(12,19)(15,20)(16,21)(18,23)(22,24);(2,4)(6,7)(10,15)(14,20)(16,22)(21,24);(5,6,8,7)(9,10,12,14,13,15,19,20)(11,16,18,21,17,22,23,24);(5,7,8,6)(9,11,13,17)(10,16,15,22)(12,18,19,23)(14,21,20,24);(10,15)(11,17)(14,20)(18,23)Polycyclic presentation:
2^9|c^2=ghi,d^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72226901, #72226911, #72226925 |
| maximal normal subgroup of | #72226901, #72226911, #72226925 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372648);