C₂×#7360 #149589

order	512
nilpotency class	5
isoclinism	isoclinic class of #1412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7360

Subgroups

	subgroup	G/H
center	C₂×C₄	#354
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#423	C₂³
socle	C₂²	#1319
prime radical	C₂⁴	#100
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

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

normal subgroups only

This group has 380 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);(3,4,6,11,7,12,21,30,8,13,22,31,23,10,20,28)(5,14,16,32,18,9,24,25,15,26,33,29,17,27,19,34);(3,5,8,15)(4,9)(6,19,22,24)(7,18,23,17)(10,14)(11,29)(12,26)(13,27)(16,20,33,21)(25,28)(30,34)(31,32)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56785);