C₂²×#563 #62690

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

Aliases

direct product

C₂²×#563

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶
prime radical	C₂³×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂² ▷ C₁

3619 conjugacy classes of subgroups; only the 2887 normal subgroups are shown.

normal subgroups only

This group has 3619 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	A₄×#563
maximal subgroup of	C₂×#8507, #187613, #187616, …
maximal normal subgroup of	C₂×#8507, #187613, #187616, …
maximal quotient of	C₂²×#2378, C₂²×#2376, C₂²×#2383, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55678);