Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437629 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2403 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
6237 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)(3,7)(4,8)(5,9)(6,10)(11,18)(12,19)(13,17)(14,21)(15,22)(16,23)(20,25)(24,29)(26,28)(27,31)(30,32);(1,3)(2,7)(4,11)(5,12)(6,13)(8,20)(9,19)(10,17)(14,24)(15,25)(16,26)(18,22)(21,30)(23,28)(27,32)(29,31);(5,16)(7,17)(9,23)(12,26)(14,27)(18,20)(21,31)(24,32);(1,4)(2,8)(3,11)(5,14)(6,15)(7,18)(9,21)(10,22)(12,24)(13,25)(16,27)(17,20)(19,29)(23,31)(26,32)(28,30);(1,5,6,16)(2,9,10,23)(3,12,13,26)(4,14,15,27)(7,19,17,28)(8,21,22,31)(11,24,25,32)(18,29,20,30)Polycyclic presentation:
2^9|e^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,g^b=gi,e^c=ei,f^d=fi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72213423, #72213523, #72213574, … |
| maximal normal subgroup of | #72213423, #72213523, #72213574, … |
| maximal quotient of | #72241292, #72241325, #72241327, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344826);