Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2965 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,11)(3,9,13,15)(4,10,5,8)(7,14,16,12)(17,18,22,26,23,19,21,25)(20,27,30,31,28,24,29,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,9)(10,15)(11,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,28)(29,32)(30,31);(1,4,6,5)(2,8,11,10)(3,12,13,14)(7,9,16,15)(17,20,22,30,23,28,21,29)(18,24,26,32,19,27,25,31);(1,3)(2,9)(4,14)(5,12)(6,13)(7,8)(10,16)(11,15)(20,28)(24,27)(29,30)(31,32);(1,4,6,5)(2,10,11,8)(3,12,13,14)(7,15,16,9)(17,21,23,22)(18,25,19,26)(20,29,28,30)(24,31,27,32)Polycyclic presentation:
2^9|a^2=g,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,f^a=fh,f^b=fh,d^c=dh,e^d=ehi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72216888, #72287820, #72319674, … |
| maximal normal subgroup of | #72216888, #72287820, #72319674, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253085);