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 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
6347 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)(11,20)(12,18)(13,19)(14,21)(15,22)(16,24)(17,26)(23,27)(25,30)(28,31)(29,32);(1,3)(2,7)(4,11)(5,12)(6,13)(8,20)(9,23)(10,19)(14,25)(15,26)(16,27)(17,22)(18,24)(21,29)(28,32)(30,31);(1,4)(2,8)(3,11)(5,14)(6,15)(7,17)(9,21)(10,22)(12,25)(13,26)(16,28)(18,29)(19,20)(23,30)(24,31)(27,32);(1,5)(2,9)(3,12)(4,14)(6,16)(7,18)(8,21)(10,24)(11,25)(13,27)(15,28)(17,29)(19,23)(20,30)(22,31)(26,32);(4,15)(8,22)(11,26)(14,28)(17,20)(21,31)(25,32)(29,30)Polycyclic presentation:
2^9|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 | #72210453, #72210507, #72210561, … |
| maximal normal subgroup of | #72210453, #72210507, #72210561, … |
| maximal quotient of | #72233069, #72233348, #72234645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344825);