Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499867 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2004 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 2004 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,15)(8,13)(9,12)(10,16)(11,17)(14,19)(18,20);(1,2)(3,4)(5,7,10,9)(6,11,16,20)(8,18,19,17)(12,13,15,14);(2,4)(6,12)(7,9)(11,13)(14,20)(15,16)(17,18);(6,13)(7,9)(8,19)(11,12)(14,16)(15,20)Polycyclic presentation:
2^9|b^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fh,e^c=eh,f^c=fi,e^d=ei,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72228001, #72228006, #72228008, … |
| maximal normal subgroup of | #72228001, #72228006, #72228008, … |
| maximal quotient of | #72268020, #72324318, #72324338 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407064);