Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437576 |
| 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 | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6081 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)(3,13)(4,10)(5,8)(6,11)(7,12)(9,15)(14,27)(16,21)(17,22)(18,25)(19,26)(20,24)(23,28)(29,32)(30,31);(2,8)(5,6)(9,13)(10,25)(11,12)(15,20)(17,18)(21,24)(22,26)(23,31)(27,32)(28,29);(1,3)(2,9)(4,14)(5,15)(6,20)(7,16)(8,21)(10,23)(11,13)(12,24)(17,28)(18,29)(19,30)(22,31)(25,27)(26,32);(1,4)(2,10)(3,14)(5,17)(6,18)(7,19)(8,22)(9,23)(11,25)(12,26)(13,27)(15,28)(16,30)(20,29)(21,31)(24,32);(1,5,7,6)(2,8,12,11)(3,15,16,20)(4,17,19,18)(9,21,24,13)(10,22,26,25)(14,28,30,29)(23,31,32,27)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,g^c=gi,f^d=fi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72233773, #72233774, #72233778, … |
| maximal normal subgroup of | #72233773, #72233774, #72233778, … |
| maximal quotient of | #72251164, #72251316, C2×#6437578, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344774);