Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3454 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)(6,8)(9,10,12,17,13,16,19,14)(11,15,18,23,20,22,24,21);(2,4)(10,14)(11,18)(12,19)(16,17)(20,24)(21,23);(5,6,7,8)(9,11,12,18,13,20,19,24)(10,15,17,23,16,22,14,21);(9,11,19,24,13,20,12,18)(10,15,14,21,16,22,17,23);(10,16)(14,17)(15,22)(21,23)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=hi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216155, #72239657, #72249399, … |
| maximal normal subgroup of | #72216155, #72239657, #72249399, … |
| maximal quotient of | #72216058, #72309070, C2×#6465638, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372834);