Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | |
2772 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)(13,14,19,16,18,22,17,21)(15,23,26,25,24,20,27,28);(6,9)(7,11)(14,16)(15,24)(17,19)(20,25)(21,22)(23,28);(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,28)(17,27)(18,24)(19,26)(21,25)(22,23);(6,9)(10,12)(13,16,17,14,18,21,19,22)(15,25,27,23,24,28,26,20);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,16,22,21)(15,26,24,27)(20,25,23,28)Polycyclic presentation:
2^9|a^2=fi,c^2=gh,d^2=f,e^2=h,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=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249598, #72249615, #72249618, … |
| maximal normal subgroup of | #72249598, #72249615, #72249618, … |
| maximal quotient of | #72264667, #72264702, #72264808, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7102854);