#65118

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #479

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
hypercenter	C₄⋊C₈	D₅
hypocenter	C₅	#479
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 29 points:

(1,2,4,5,3)(8,12)(10,13)(14,16,17,24,18,23,26,27)(15,19,20,22,21,28,25,29);(6,8,9,12)(7,10,11,13)(16,23)(19,28)(22,29)(24,27);(2,3)(4,5)(6,7,9,11)(8,10,12,13)(14,15,18,21)(16,22,23,29)(17,25,26,20)(19,27,28,24)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10610756, #10610760, #10610761, …
maximal normal subgroup of	#10610756, #10610760, #10610761, …
maximal quotient of	#10599414, #10599416, #10599427, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 456);