Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6442597 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#535 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2408 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,11)(4,8)(5,17)(6,10)(7,14)(9,18)(12,24)(13,23)(15,21)(16,29)(19,27)(20,26)(22,28)(25,31)(30,32);(7,19)(8,21)(11,23)(22,29);(1,3,5,14,6,13,18,27)(2,7,9,11,10,19,17,23)(4,12,16,26,15,25,28,32)(8,20,22,24,21,30,29,31);(1,4)(2,8)(3,12)(5,16)(6,15)(7,20)(9,22)(10,21)(11,24)(13,25)(14,26)(17,29)(18,28)(19,30)(23,31)(27,32);(3,13)(7,19)(11,23)(12,25)(14,27)(20,30)(24,31)(26,32)Polycyclic presentation:
2^9|c^2=g,g^2=i,b^a=bf,c^a=cg,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216071, #72216095, #72216320, … |
| maximal normal subgroup of | #72216071, #72216095, #72216320, … |
| maximal quotient of | #72251122, #72251130, C2×#6442738, … |
| automorphism group of | #2166 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349934);