Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806575 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #39743 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,21)(15,24,16,20)(17,27,26,22)(19,23,25,28);(5,7)(6,9)(8,12)(10,11)(13,15,18,16)(14,20,21,24)(17,25,26,19)(22,28,27,23);(1,2,3,4)(13,16,19,17,18,15,25,26)(14,20,23,27,21,24,28,22);(6,10)(9,11)(14,21)(20,24)(22,27)(23,28);(7,12)(9,11)(13,17,25,16,18,26,19,15)(14,22,28,24,21,27,23,20)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gh,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6721648);