Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667676 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #29712 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1871 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,12)(11,14,16,15)(17,18,21,24)(19,23,20,22);(1,3)(2,5)(4,8)(6,7)(9,11,13,16)(10,14,12,15);(9,10,13,12)(11,15,16,14)(18,22)(19,20)(23,24);(2,6)(5,7);(3,8)(5,7)(9,12,13,10)(11,14,16,15)(17,19,21,20)(18,23,24,22)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72282132, #72283403, #72305299, … |
| maximal normal subgroup of | #72282132, #72283403, #72305299, … |
| maximal quotient of | #72305633, #72305749, #72316229 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6761934);