#72335964

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

35743 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(6,9)(10,12)(14,17)(18,19)(22,25)(23,27);(21,22)(23,26)(24,25)(27,28);(6,9)(10,12);(2,4)(7,11)(10,12)(23,27)(26,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,17)(18,20)(21,22)(23,28)(24,25)(26,27);(5,7,8,11)(6,10,9,12)(13,15,16,20)(14,18,17,19)(21,23)(22,26)(24,27)(25,28)

Polycyclic presentation:

2^10|e^2=gh,f^2=gj,b^a=bg,e^a=ej,e^c=ei,e^d=egh,f^a=fg,f^d=fgi,f^e=fij

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336001
maximal normal subgroup of	#72336001
maximal quotient of	#72336455

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, 5, F.7*F.8);;
gap> SetPower(coll, 6, F.7*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);