C₂²×#578 #63086

order	256
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂²×#578

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁶
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

5387 conjugacy classes of subgroups; only the 3143 normal subgroups are shown.

normal subgroups only

This group has 5387 conjugacy classes of subgroups — too many to draw. Only its 3143 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 36 points:

(1,2);(3,4);(5,6,10,14)(7,15,18,28)(8,12,19,26)(9,21,23,13)(11,22,25,17)(16,29,30,36)(20,33,32,27)(24,34,35,31);(5,7,10,18)(6,11,14,25)(8,16,19,30)(9,22,23,17)(12,24,26,35)(13,28,21,15)(20,34,32,31)(27,36,33,29);(5,8)(6,12)(7,16)(9,20)(10,19)(11,24)(13,27)(14,26)(15,29)(17,31)(18,30)(21,33)(22,34)(23,32)(25,35)(28,36);(8,19)(12,26)(16,30)(20,32)(24,35)(27,33)(29,36)(31,34)

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	A₄×#578, C₂²×#578⋊C₃, #11710892
maximal subgroup of	#359612, #359616, #359677, …
maximal normal subgroup of	#359612, #359616, #359677, …
maximal quotient of	C₂²×#2898, C₂²×#2913, C₂²×#2951, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 56074);