Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
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 | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2233 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(13,14)(15,22)(16,20)(17,27)(18,21)(19,25)(23,26)(24,28);(5,7,9,8)(6,10,11,12)(13,15,18,26)(14,19,21,28)(16,25,17,24)(20,23,27,22);(5,8,9,7)(6,10,11,12)(13,14)(15,19)(16,27)(17,20)(18,21)(22,25)(23,24)(26,28);(13,14)(15,23)(16,20)(17,27)(18,21)(19,24)(22,26)(25,28);(13,16,18,17)(14,20,21,27)(15,24,26,25)(19,22,28,23)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,d^b=df,e^b=ei,f^b=fi,e^c=ei,f^d=fi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72307281, #72307428, #72307642, … |
| maximal normal subgroup of | #72307281, #72307428, #72307642, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287733);