Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93181 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1304 |
| commutator | C23×C4 | C42 |
| Frattini, cosocle | C22×#124 | C22 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C22×#124 ▷ C24 ▷ C22 ▷ C1 | |
435 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 435 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2,4,7)(3,9,10,15)(5,8,13,16)(6,11,14,12)(17,18,21,25,22,26,37,44)(19,27,29,45,30,41,35,47)(20,31,33,43,34,24,36,42)(23,38,39,32,40,46,48,28);(1,3)(2,6,8,15)(4,12,13,10)(5,11)(7,9)(14,16)(17,19)(18,23,26,40)(20,32,34,28)(21,35)(22,30)(24,41)(25,39,44,48)(27,31)(29,37)(33,38,36,46)(42,47)(43,45)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=h,d^2=h,b^a=bc,c^a=cf,e^a=egh,f^a=fh,g^a=gi,c^b=cg,d^b=dfh,f^b=fi,g^b=ghi,d^c=dh,e^c=ehi,e^d=ei32 conjugacy classes — view the full character table.
| maximal subgroup of | #72227933, #72227935, #72235369, … |
| maximal normal subgroup of | #72227933, #72227935, #72235369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380);