Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,30)(21,23)(22,29)(24,26)(27,31)(28,32);(9,11)(10,13)(12,15)(14,16)(17,19,26,28,21,27,20,29)(18,22,30,31,23,32,24,25);(1,2,3,4)(11,15)(13,16)(18,23)(19,27)(24,30)(28,29);(6,8)(19,28)(20,26)(22,31)(24,30)(25,32)(27,29)Polycyclic presentation:
2^9|a^2=gi,b^2=ei,c^2=fi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72307769, #72307794, #72307902, … |
| maximal normal subgroup of | #72307769, #72307794, #72307902, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341047);