Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,3,4)(5,6)(7,8)(9,10,13,15)(11,16,12,14)(17,18,22,25,23,20,21,26)(19,27,29,24,31,28,30,32);(2,4)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,32)(21,29)(22,30)(23,31)(25,27)(26,28);(2,4)(9,12,13,11)(10,14,15,16)(17,20)(18,23)(19,28)(21,26)(22,25)(24,29)(27,31)(30,32);(6,8)(17,21,23,22)(18,25,20,26)(19,29,31,30)(24,27,32,28)Polycyclic presentation:
2^9|a^2=ei,b^2=ef,c^2=ef,d^2=i,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 | #72237235, #72237269, #72239617, … |
| maximal normal subgroup of | #72237235, #72237269, #72239617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 306395);