#72320195

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁵
commutator, socle	C₂³	C₂³×C₄²
prime radical	C₂×#525	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

5167 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=f,d^2=gi,f^2=i,h^2=j,b^a=bj,d^a=dj,d^b=dg,d^c=dgi,e^d=ei

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320264
maximal normal subgroup of	#72320264
maximal quotient of	#72322207

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