C₂²×#350 #33338

order	256
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) order 24576
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#350

Subgroups

	subgroup	G/H
center	C₂²×C₈	D₄
commutator	C₄	C₂³×C₈
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂³	C₂²⋊C₈
prime radical	C₂²×#41	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

587 conjugacy classes of subgroups; only the 271 normal subgroups are shown.

normal subgroups only

This group has 587 conjugacy classes of subgroups — too many to draw. Only its 271 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 — 4 generators on 36 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	A₄×#350, S₄×#350, #72343077
maximal subgroup of	#146384, #146389, #146412, …
maximal normal subgroup of	#146384, #146389, #146412, …
maximal quotient of	C₂²×#1315, C₂×#11422, C₂²×#1323, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26326);