C₈×#368 #129148

order	512
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₈×#368

Subgroups

	subgroup	G/H
center	C₄×C₈	D₈
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×#107
prime radical	C₂³	C₄×Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

262 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,5,4,6,7,8);(9,10,13,18,15,19,14,29)(11,20,23,35,25,36,24,39)(12,26,27,33,28,40,21,17)(16,30,31,38,32,37,34,22);(9,11,21,37,14,24,28,38,15,25,27,30,13,23,12,22)(10,16,26,35,18,31,33,36,19,32,40,39,29,34,17,20)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×#2179, C₈×#1426, C₈×#1419, …
maximal normal subgroup of	C₈×#2179, C₈×#1426, C₈×#1419, …
maximal quotient of	C₂×C₈×#368

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36344);