Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
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×#3347 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(14,20)(15,22)(17,19)(21,28)(23,24)(25,27);(5,7)(6,9)(8,12)(10,11)(13,14)(15,23)(16,19)(17,26)(18,20)(21,25)(22,28)(24,27);(13,15,17,27,18,25,19,22)(14,21,26,28,20,23,16,24);(7,12)(9,11)(13,16)(14,17)(15,24)(18,26)(19,20)(21,27)(22,23)(25,28);(1,3,4,2)(6,10)(9,11)(13,17,18,19)(14,16,20,26)(15,22,25,27)(21,28,23,24)Polycyclic presentation:
2^9|a^2=h,c^2=gi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^b=ei,e^c=ei,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6310663);