Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7535048 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3549 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | #60045 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1373 conjugacy classes of subgroups; only the 428 normal subgroups are shown.
This group has 1373 conjugacy classes of subgroups — too many to draw. Only its 428 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,12)(4,17)(5,9)(6,11)(7,14)(8,20)(10,21)(13,22)(15,24)(16,23)(18,28)(19,27)(25,30)(26,31)(29,32);(1,3,5,14,6,16,21,15)(2,7,10,12,11,24,9,23)(4,18,20,13,19,30,31,29)(8,25,17,22,26,28,27,32);(1,4,6,19)(2,8)(3,13)(5,20,21,31)(7,22,24,32)(9,27)(10,17)(11,26)(12,28,23,25)(14,30)(15,18)(16,29);(2,9,11,10)(4,19)(7,12,24,23)(8,17,26,27)(13,29)(18,30)(20,31)(22,25,32,28);(2,9,11,10)(4,19)(7,23,24,12)(8,17,26,27)(13,29)(18,30)(20,31)(22,28,32,25)Polycyclic presentation:
2^9|b^2=f,c^2=fgh,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| Fitting subgroup of | #72227427 |
| maximal subgroup of | #72217950, #72227427, #72227436, … |
| maximal normal subgroup of | #72217950, #72227427, #72227436, … |
| maximal quotient of | #72216206, #72216224, #72216230, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529606);