Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2055 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)(7,9)(8,10)(11,12)(13,14,19,21,18,24,17,23)(15,20,26,27,16,22,25,28);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,16)(14,20,24,22)(17,25,19,26)(21,27,23,28);(5,7,8,12)(6,9,10,11)(14,21)(17,19)(20,27)(22,28)(23,24)(25,26);(1,3,4,2)(7,12)(9,11)(13,16,18,15)(14,22,24,20)(17,26,19,25)(21,28,23,27);(6,10)(9,11)(15,16)(20,22)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=h,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
| maximal subgroup of | #72280269, #72280659, #72280757, … |
| maximal normal subgroup of | #72280269, #72280659, #72280757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6971493);