C₂×#7459 #151028

order	512
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×#7459

Subgroups

	subgroup	G/H
center	C₂×C₁₆	D₈
commutator	C₈	C₂²×C₁₆
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂²	#1370
prime radical	C₂³	#364
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

174 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216243, C₂×#151634, #72216267, …
maximal normal subgroup of	#72216243, C₂×#151634, #72216267, …
maximal quotient of	#72225400

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58224);