Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2397 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2866 conjugacy classes of subgroups; only the 794 normal subgroups are shown.
This group has 2866 conjugacy classes of subgroups — too many to draw. Only its 794 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(7,11)(10,12)(14,17)(18,20)(21,22,25,26)(23,27,29,30)(24,32,33,28)(31,36,35,34);(6,9)(10,12)(14,17)(15,19)(22,26)(23,29)(28,32)(31,35);(5,6,8,9)(7,10,11,12)(13,14,16,17)(15,18,19,20)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,33)(34,35);(5,7)(6,10)(8,11)(9,12)(13,15,16,19)(14,18,17,20)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,36)(33,35);(21,24,25,33)(22,28,26,32)(23,31,29,35)(27,34,30,36)Polycyclic presentation:
2^9|a^2=i,c^2=fh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dgh,e^a=ef,c^b=ch,d^b=dg,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72293251, #72293260 |
| maximal normal subgroup of | #72293251, #72293260 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 489420);