Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2040 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)(7,8)(9,11)(10,12)(13,14,18,22,17,19,16,26)(15,23,25,21,24,28,27,20);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(14,19)(15,24)(22,26)(25,27);(2,4)(5,8,9,10)(6,7,11,12)(13,15)(14,20)(16,25)(17,24)(18,27)(19,21)(22,28)(23,26);(6,11)(7,12)(13,15)(14,21)(16,25)(17,24)(18,27)(19,20)(22,23)(26,28);(13,16,17,18)(14,22,19,26)(15,25,24,27)(20,28,21,23)Polycyclic presentation:
2^9|a^2=g,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72285178 |
| maximal normal subgroup of | #72285178 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410613);