Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(6,9)(7,11)(14,17)(15,19);(5,6,8,9)(7,12,11,10)(13,14)(15,18)(16,17)(19,20)(22,25)(23,26)(27,28);(13,15,16,19)(14,18,17,20)(21,22,26,28,24,27,23,25);(5,7,8,11)(6,10,9,12)(14,17)(18,20)(22,27)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=fgi,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72320024, #72320096, #72320384 |
| maximal normal subgroup of | #72320024, #72320096, #72320384 |
| maximal quotient of | #72321080 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294184);