Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| 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 | |
2934 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)(12,15)(16,20)(17,21)(19,22)(23,24);(2,4)(6,7);(5,6,8,7)(9,10,15,21,13,17,12,14)(11,16,22,24,18,23,19,20);(5,7,8,6)(9,11)(10,16)(12,19)(13,18)(14,20)(15,22)(17,23)(21,24);(10,17)(11,18)(14,21)(19,22)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=dh,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249920, #72249947, #72257009, … |
| maximal normal subgroup of | #72249920, #72249947, #72257009, … |
| maximal quotient of | #72264291 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372643);