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 | #61431 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2965 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,10,11,8)(4,6,13,9)(12,15,16,14)(17,18,21,25,22,19,23,27)(20,24,29,31,28,26,30,32);(2,6)(4,13)(7,9)(8,14)(10,15)(12,16);(2,7)(3,11)(6,9)(12,16)(17,19,21,27,22,18,23,25)(20,26,29,32,28,24,30,31);(1,3)(2,8)(4,12)(5,11)(6,14)(7,10)(9,15)(13,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32);(3,11)(8,10)(12,16)(14,15)(20,28)(24,26)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,c^2=gh,f^2=h,g^2=i,b^a=bf,c^a=ch,d^a=dh,f^a=fh,f^b=fh,d^c=dh,e^d=ehi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72216788, #72216810, #72216884, … |
| maximal normal subgroup of | #72216788, #72216810, #72216884, … |
| maximal quotient of | C2×#6345910, #72318337, #72318344, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253106);