Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11867 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1519 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C2×#469 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
398 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,6,7,8)(9,10,13,17,16,21,12,19,14,22,29,33,15,20,28,32)(11,18,24,35,27,37,39,40,25,38,30,31,26,36,23,34);(1,2)(3,6)(4,8)(5,7)(10,17)(12,13)(15,16)(18,31)(19,20)(21,32)(22,33)(23,30)(24,39)(26,27)(28,29)(34,36)(35,38)(37,40);(1,3,5,4)(2,6,7,8)(9,11)(10,18)(12,23)(13,30)(14,25)(15,26)(16,27)(17,31)(19,34)(20,36)(21,37)(22,38)(24,29)(28,39)(32,40)(33,35)Polycyclic presentation:
2^9|a^2=e,c^2=gh,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cf,d^a=df,f^a=fi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72265302, #72265452, #72265457, … |
| maximal normal subgroup of | #72265302, #72265452, #72265457, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31637);