Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,21)(20,22)(23,24);(2,5)(6,8)(17,19)(18,21)(20,23)(22,24);(9,11)(10,14)(12,15)(13,16)(18,22)(19,23);(10,15)(11,16)(18,22)(21,24);(9,12,13,14)(10,11,15,16);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72234893, #72234894, #72234896, … |
| maximal normal subgroup of | #72234893, #72234894, #72234896, … |
| maximal quotient of | #72262085, C2×#7633633, #72303648, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540829);