C₂²×#560 #62687

order	256
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂²×#560

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂² ▷ C₁

4803 conjugacy classes of subgroups; only the 2887 normal subgroups are shown.

normal subgroups only

This group has 4803 conjugacy classes of subgroups — too many to draw. Only its 2887 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 20 points:

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72234943, #72235179, #72235272
Frattini subgroup of	#72234507, #72234526, #72234943, …
Fitting subgroup of	A₄×#560, C₂²×#560⋊C₃, #11710644, …
maximal subgroup of	#188063, #188064, #188065, …
maximal normal subgroup of	#188063, #188064, #188065, …
maximal quotient of	C₂²×#2375, C₂²×#2377, C₂²×#2376, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55675);