#64985

order	320
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₄×C₅×C₈ ⋊ C₂
Sylow	C₅ ⋊ #495

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₄×C₅×C₈	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂×D₄₀	C₂
hypercenter	C₄×C₈	D₅
hypocenter	C₅	#495
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 25 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10597958, #10598053, #10599852, …
maximal normal subgroup of	#10597958, #10598053, #10599852, …
maximal quotient of	#10598851, #10598852, #10598853, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 323);