#65285

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #485

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#117	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂³×C₅	D₄
hypercenter	#117	D₅
hypocenter	C₅	#485
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 160

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10610578, #10610579, #10610580, …
maximal normal subgroup of	#10610578, #10610579, #10610580, …
maximal quotient of	#10600251, #10600252, #10600257, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 623);