Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #52117 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1923 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,19,24,18,22,16,21)(15,17,23,28,25,20,26,27);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,26)(17,22)(18,25)(19,23)(21,28)(24,27);(1,2,3,4)(6,10)(9,11)(13,16,18,19)(14,21,22,24)(15,23,25,26)(17,28,20,27);(7,12)(9,11)(14,22)(15,25)(21,24)(23,26);(6,10)(9,11)(13,17,19,28,18,20,16,27)(14,23,24,25,22,26,21,15)Polycyclic presentation:
2^9|a^2=h,c^2=fg,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ci,d^b=df,e^b=ei,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6310972);