#65529

order	320
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #410

Subgroups

	subgroup	G/H
center	C₄	C₂²⋊Dic₅
commutator	C₂²×C₅	C₂²×C₄
Frattini	C₂³	C₂²×D₅
Fitting, prime radical	C₂×C₅×#41	C₂
socle	C₂×C₅	C₂×C₂²⋊C₄
cosocle	C₂³×C₅	C₂³
hypercenter	C₂×#41	D₅
hypocenter	C₅	#410
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 21 points:

(1,2,4,5,3)(6,9)(7,12)(8,14)(10,17)(11,18)(13,16)(15,21)(19,20);(6,8,10,15)(7,11,13,19)(9,14,17,21)(12,18,16,20);(2,3)(4,5)(6,7)(8,11)(9,16,17,12)(10,13)(14,20,21,18)(15,19)

Polycyclic presentation:

2^6,5|a^2=d,b^2=f,c^a=ce,e^a=ef,g^a=g^4,d^c=df

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598182, #10598183, #10598187, …
maximal normal subgroup of	#10598182, #10598183, #10598187, …
maximal quotient of	#10602364, #10602365, #10602385, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 867);