Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| 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 | #34736 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14,18,22)(15,23,25,28)(16,21,17,20)(19,26,27,24);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11);(2,4)(13,15)(14,19)(16,24)(17,26)(18,25)(20,23)(21,28)(22,27);(6,10)(7,12)(13,15)(14,19)(16,24)(17,26)(18,25)(20,23)(21,28)(22,27);(13,16,18,17)(14,20,22,21)(15,24,25,26)(19,23,27,28)Polycyclic presentation:
2^9|a^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72237210, #72249628, #72264307, … |
| maximal normal subgroup of | #72237210, #72249628, #72264307, … |
| maximal quotient of | #72238925, #72265104, #72265175, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407862);