C₂²×#443 #60177

order	256
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) order 196608
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×#443

Subgroups

	subgroup	G/H
center	C₂²×C₈	D₄
commutator	C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₄	C₂⁵
socle	C₂³	C₄×D₄
prime radical	C₂²×#135	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

1375 conjugacy classes of subgroups; only the 783 normal subgroups are shown.

normal subgroups only

This group has 1375 conjugacy classes of subgroups — too many to draw. Only its 783 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 — 5 generators on 20 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	A₄×#443, S₄×#443, #72342727, …
maximal subgroup of	#129508, #129509, #146729, …
maximal normal subgroup of	#129508, #129509, #146729, …
maximal quotient of	C₂²×C₈×D₈, C₂²×C₈×SD₁₆, C₂²×C₈×Q₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53165);