#72279587

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×AΣL(1,𝔽₁₆)
commutator	C₂²×D₄	C₂³×C₄
Frattini, cosocle	C₂×#127	C₂⁴
prime radical	C₂×#23906	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂³×#127 ◁ G
LECS	G ▷ C₂×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

5739 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 20 points:

(5,6)(7,14,15,19,16,20,17,10)(8,12)(9,11)(13,18);(1,2)(3,4);(2,4)(5,7,9,16)(6,10,11,19)(8,17,18,15)(12,14,13,20);(6,11)(10,19)(12,13)(14,20)

Polycyclic presentation:

2^10|a^2=g,c^2=j,g^2=ij,h^2=j,c^a=cf,c^b=ce,d^a=dj,f^a=fh,g^c=ghj,g^f=gij,h^a=hi,h^c=hj,h^f=hj,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279660, #72279773, #72279879, …
maximal normal subgroup of	#72279660, #72279773, #72279879, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);