Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| 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 | |
2788 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)(13,14,17,20,18,21,27,16)(15,22,25,26,24,19,28,23);(6,9)(7,11)(14,16)(17,27)(19,26)(20,21)(22,23)(25,28);(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,15,18,24)(14,19,21,22)(16,26,20,23)(17,25,27,28);(6,9)(10,12)(13,16,17,14,18,20,27,21)(15,23,25,22,24,26,28,19);(5,7,8,11)(6,10,9,12)(15,24)(19,22)(23,26)(25,28)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72249595, #72249618, #72261272, … |
| maximal normal subgroup of | #72249595, #72249618, #72261272, … |
| maximal quotient of | #72264429, #72264515, #72265229, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7102844);