C₂²×#568 #63074

order	256
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂²×#568

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁴
commutator	C₂	C₂⁵×C₄
Frattini, cosocle	C₄	C₂⁶
socle	C₂³	C₂³×C₄
prime radical	C₂³×#41	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

5559 conjugacy classes of subgroups; only the 5287 normal subgroups are shown.

normal subgroups only

This group has 5559 conjugacy classes of subgroups — too many to draw. Only its 5287 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:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	A₄×#568, C₂²×#568⋊C₃, #11710869
maximal subgroup of	#353297, #353299, #353322, …
maximal normal subgroup of	#353297, #353299, #353322, …
maximal quotient of	C₂²×#2873, C₂²×#2875, C₂×C₄×#568, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 56062);