Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5821 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,18)(19,22)(20,21)(23,24);(1,3,5,4)(2,7,6,8)(10,15)(14,16)(18,21)(22,24);(9,12,13,11)(10,16,15,14)(17,18)(19,22)(20,21)(23,24);(1,4,5,3)(2,8,6,7)(18,21)(22,24);(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=h,e^2=g,f^2=i,b^a=bg,c^a=ch,c^b=ci,e^b=ei,d^c=di,e^d=ei140 conjugacy classes — view the full character table.
| maximal subgroup of | #72279634, C2×#7630412 |
| maximal normal subgroup of | #72279634, C2×#7630412 |
| maximal quotient of | #72280386, #72280503, C2×#7630412 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537608);