Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499780 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,25,29)(23,31,32,27)(24,30,26,28)(33,36,34,35);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23)(22,27)(24,33)(25,32)(26,34)(28,36)(29,31)(30,35);(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(22,28)(23,32)(24,26)(27,35)(29,30)(31,36);(8,16)(11,20)(13,15)(17,19)(21,24,25,26)(22,28,29,30)(23,33,32,34)(27,36,31,35);(7,14)(10,18)(13,15)(17,19)(23,32)(27,31)(33,34)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72314763, #72314765, #72314794, … |
| maximal normal subgroup of | #72314763, #72314765, #72314794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6415922);