C₂×#8509 #187610

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#8509

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2094 conjugacy classes of subgroups; only the 362 normal subgroups are shown.

normal subgroups only

This group has 2094 conjugacy classes of subgroups — too many to draw. Only its 362 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) = 512

A minimal faithful permutation representation — 4 generators on 38 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94806);