Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437428 |
| 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 | #23430 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(14,18)(16,23)(19,25)(20,24)(22,27)(26,28);(7,8)(10,12)(13,14,17,20)(15,19,21,26)(16,24,23,18)(22,28,27,25);(1,2,3,4)(5,7,9,8)(6,10,11,12);(5,8,9,7)(6,10,11,12)(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,27)(24,28);(15,21)(19,26)(22,27)(25,28)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72249595, #72249615, #72250828 |
| maximal normal subgroup of | #72249595, #72249615, #72250828 |
| maximal quotient of | #72264500, #72307859, #72308775, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370070);