Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500807 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
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 | |
1075 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1075 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(4,8)(5,17)(9,20)(10,27)(11,22)(12,29)(13,23)(18,21)(25,30)(26,31)(28,32);(1,2,8,22,6,11,4,7)(3,12,23,32,15,28,13,29)(5,18,24,21,17,10,16,27)(9,25,31,19,26,30,20,14);(1,3)(2,9)(4,13)(5,19)(6,15)(7,20)(8,23)(10,28)(11,26)(12,18)(14,17)(16,25)(21,32)(22,31)(24,30)(27,29);(1,2,8,22,6,11,4,7)(3,9,23,31,15,26,13,20)(5,10,24,27,17,18,16,21)(12,25,32,19,28,30,29,14)Polycyclic presentation:
2^9|b^2=ei,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72236850, #72236900, #72236997, … |
| maximal normal subgroup of | #72236900, #72236997, #72237526, … |
| maximal quotient of | #72311838, #72311842, C2×#500810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408006);