Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3151 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,8,7)(10,16)(17,18)(19,20)(21,23);(6,7)(9,10,14,16)(11,21,12,23)(13,20,15,19)(17,24,18,22);(1,2,3,4)(5,6,8,7)(9,11,14,12)(10,17,16,18)(13,22,15,24)(19,21,20,23);(5,7,8,6)(9,12,14,11)(10,18,16,17)(13,24,15,22)(19,23,20,21);(9,13,14,15)(10,19,16,20)(11,22,12,24)(17,21,18,23)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72307351, #72307365, #72307460, … |
| maximal normal subgroup of | #72307351, #72307365, #72307460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351369);