Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62283 |
| 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 | C2×#3192 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,14,19,22)(15,20,25,28)(16,17,21,18)(23,24,27,26);(1,2,4,3)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,23)(17,24)(18,26)(19,25)(21,27)(22,28);(1,3,4,2)(13,16,18,14,19,21,17,22)(15,23,26,20,25,27,24,28);(7,12)(9,11)(13,17,19,18)(14,21,22,16)(15,24,25,26)(20,27,28,23);(6,10)(9,11)(15,25)(20,28)(23,27)(24,26)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=ghi,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72282404, #72282422, #72282446, … |
| maximal normal subgroup of | #72282404, #72282422, #72282446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6303307);