C₂×#7368 #149617

order	512
nilpotency class	5
isoclinism	isoclinic class of #1414
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7368

Subgroups

	subgroup	G/H
center	C₂×C₄	#354
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#423	C₂³
socle	C₂²	#1324
prime radical	C₂⁴	#101
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#423 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

296 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 82 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);(3,4,6,12,8,13,22,30,9,14,21,31,7,11,23,28)(5,15,16,29,18,10,24,25,19,26,32,34,17,27,20,33);(3,5,7,17,9,19,8,18)(4,10,13,26,14,27,11,15)(6,20,22,16,21,24,23,32)(12,29,28,33,31,34,30,25)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56813);