Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,14)(15,17)(16,18)(19,20)(21,22,25,27,26,23,24,28);(2,4)(6,9)(10,12)(13,15,16,19)(14,17,18,20)(22,27)(23,28)(24,25);(2,4)(21,23)(22,26)(24,28)(25,27);(5,7,8,11)(6,10,9,12)(14,18)(17,20)Polycyclic presentation:
2^9|a^2=ei,b^2=h,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237292, #72239610, #72239634, … |
| maximal normal subgroup of | #72237292, #72239610, #72239634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 314897);