#65286

order	320
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #405

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅
commutator	C₂×C₅	C₂×C₄²
Fitting	C₅×#117	C₂
socle	C₂²×C₅	C₂²×C₄
prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	#117	D₅
hypocenter	C₅	#405
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #117

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

Polycyclic presentation:

2^6,5|a^2=d,c^2=e,d^2=f,g^a=g^4,c^b=cf

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598728, #10598729, #10599490, …
maximal normal subgroup of	#10598728, #10598729, #10599490, …
maximal quotient of	#10600227, #10600228, #10600229, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 624);