Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2654 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)(7,11)(10,12)(13,14,18,23,19,16,17,22)(15,21,24,26,25,27,28,20);(6,9)(7,11)(13,15)(14,20)(16,26)(17,24)(18,28)(19,25)(21,22)(23,27);(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,15)(14,21)(16,27)(17,28)(18,24)(19,25)(20,22)(23,26);(6,9)(10,12)(13,16,18,22,19,14,17,23)(15,21,24,26,25,27,28,20);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,22,16,23)(15,24,25,28)(20,21,26,27)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72249607, #72249615, #72249631, … |
| maximal normal subgroup of | #72249607, #72249615, #72249631, … |
| maximal quotient of | #72264725, #72264827, #72265071, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7102842);