Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #347042 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
3878 conjugacy classes of subgroups; only the 196 normal subgroups are shown.
This group has 3878 conjugacy classes of subgroups — too many to draw. Only its 196 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,6)(7,12)(8,13)(14,16);(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16);(3,9)(6,12)(10,15)(13,16);(3,10)(6,13)(9,15)(12,16)Polycyclic presentation:
2^9|b^a=be,c^a=cf,d^a=dg,f^b=fh,g^b=gi,e^c=eh,e^d=ei59 conjugacy classes — view the full character table.
| maximal subgroup of | #72209641, #72209642, B(5,𝔽2), … |
| maximal normal subgroup of | B(5,𝔽2), #72209835, #72209882, … |
| maximal quotient of | #72209757, #72209767, #72210498, … |
| automorphism group of | C2×C4⋊C4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 254238);