C₂×#7505 #151849

order	512
nilpotency class	5
isoclinism	isoclinic class of #1412
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#7505

Subgroups

	subgroup	G/H
center, socle	C₂³	#354
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄⋊Q₈	C₂³
prime radical	C₂⁴	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

642 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 38
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 38 points:

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

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, 59045);