#72249462

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 15781 conjugacy classes of subgroups; only the 31817 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 7 generators on 28 points:

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

Polycyclic presentation:

2^10|a^2=i,b^2=i,b^a=bi,c^a=ci,e^a=eh,e^c=ej,e^d=ei,f^b=fhj,g^b=gi,g^e=gj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249567
maximal normal subgroup of	#72249567
maximal quotient of	#72326901, #72327934, #72329191, …

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