Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62283 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3199 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(14,18)(15,21)(19,23)(20,24)(22,28)(25,27);(1,2,4,3)(5,7)(6,9)(8,12)(10,11)(13,14,17,18)(15,21,23,19)(16,24,26,20)(22,27,25,28);(1,3,4,2)(13,15,14,19,17,23,18,21)(16,25,20,28,26,22,24,27);(7,12)(9,11)(13,16,17,26)(14,20,18,24)(15,22,23,25)(19,27,21,28);(6,10)(9,11)(13,14,17,18)(15,19,23,21)(16,24,26,20)(22,28,25,27)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=ghi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,d^c=di,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72311834, #72311850, #72311876, … |
| maximal normal subgroup of | #72311834, #72311850, #72311876, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6303355);