Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | |
2738 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)(13,14,18,16,19,23,17,22)(15,21,25,24,26,27,28,20);(6,9)(7,11)(13,15)(14,20)(16,27)(17,25)(18,28)(19,26)(21,22)(23,24);(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,15)(14,21)(16,24)(17,28)(18,25)(19,26)(20,22)(23,27);(6,9)(10,12)(13,16,17,14,19,22,18,23)(15,24,28,21,26,20,25,27);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,22,23,16)(15,25,26,28)(20,21,24,27)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=fi,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=dh,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249595, #72249607, #72249631, … |
| maximal normal subgroup of | #72249595, #72249607, #72249631, … |
| maximal quotient of | #72264552, #72264597, #72264618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7102829);