#72334400

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁵
prime radical	C₂³×D₄²	C₂
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

62547 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^10|a^2=g,e^2=fgij,b^a=bgh,c^a=cf,c^b=cj,d^a=dhi,d^c=dj,e^a=efij,e^b=efij,e^c=ef,e^d=ehij

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334419, #72334422, #72334424, …
maximal normal subgroup of	#72334419, #72334422, #72334424, …
maximal quotient of	C₂×#72334400, #72337401

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