#72299697

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁵
commutator	C₂³	C₂³×C₄²
prime radical	C₂²×#392	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

17979 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 20 points:

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

Polycyclic presentation:

2^10|b^2=gj,c^2=h,d^2=gj,e^2=fh,b^a=bg,d^c=dghi,e^a=ef,e^b=ef,e^c=ef,e^d=ef

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72299861
maximal subgroup of	#72299698, #72299861
maximal normal subgroup of	#72299698, #72299861
maximal quotient of	#72299929

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