Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93139 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1304 |
| commutator | C25 | C42 |
| Frattini, cosocle | C22×#124 | C22 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C25 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C22×#124 ▷ C24 ▷ C22 ▷ C1 | |
418 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 418 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)(19,27)(20,31,33,24)(21,25)(22,26)(23,28,37,32)(29,45)(30,39)(34,41,35,40)(36,42)(38,46,48,44)(43,47);(1,3)(2,6,8,15)(4,12,13,10)(5,11)(7,9)(14,16)(17,19,21,29,22,30,36,47)(18,23,25,38,26,37,42,48)(20,32,34,46,33,28,35,44)(24,39,40,45,31,27,41,43)Polycyclic presentation:
2^9|a^2=d,b^2=e,e^2=h,b^a=bc,c^a=cf,e^a=eg,f^a=fh,g^a=gi,c^b=cg,d^b=df,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 | #72235366, #72235374, #72235375, … |
| maximal normal subgroup of | #72235366, #72235374, #72235375, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337);