Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| 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 | #34727 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1965 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,8)(9,11)(10,12)(13,14,18,22)(15,19,17,20)(16,24,26,27)(21,25,28,23);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,18,17)(14,19,22,20)(16,23,26,25)(21,27,28,24);(2,4)(14,20)(15,17)(19,22)(21,24)(23,25)(27,28);(6,11)(7,12)(13,16,18,26)(14,21,22,28)(15,23,17,25)(19,27,20,24);(5,8,9,10)(6,7,11,12)(13,15,18,17)(14,19,22,20)(16,25,26,23)(21,24,28,27)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^c=ei,g^c=gi,e^d=ei,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409886);