Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500803 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24260 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1041 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,3)(4,15,19,27)(5,6,13,9)(7,11,22,14)(10,25,12,18)(16,31,26,29)(17,32,23,30)(20,21,28,24);(1,3,6,13,8,2,9,5)(4,12,17,26,19,10,23,16)(7,11,21,20,22,14,24,28)(15,29,30,18,27,31,32,25);(1,4)(2,10)(3,12)(5,16)(6,17)(7,18)(8,19)(9,23)(11,15)(13,26)(14,27)(20,30)(21,31)(22,25)(24,29)(28,32);(1,5,6,3,8,13,9,2)(4,16,17,12,19,26,23,10)(7,20,21,14,22,28,24,11)(15,18,30,31,27,25,32,29)Polycyclic presentation:
2^9|a^2=i,b^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,d^b=di,f^b=fh,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72236831, #72236887, #72236890, … |
| maximal normal subgroup of | #72236890, #72237519, #72237521, … |
| maximal quotient of | #72311949, #72312136, #72312358, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408000);