Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2907 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)(9,10,15,16,14,12,13,18)(11,19,20,24,21,17,22,23);(5,6)(7,8)(10,16)(11,20)(12,18)(13,15)(17,19)(21,22);(1,2,3,4)(6,8)(9,11,15,20,14,21,13,22)(10,17,16,23,12,19,18,24);(6,8)(9,12,13,16,14,10,15,18)(11,19,22,23,21,17,20,24);(11,21)(17,19)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307673, #72307781, #72307847, … |
| maximal normal subgroup of | #72307673, #72307781, #72307847, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351567);