C₂×#9004 #194476

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#9004

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

1218 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 1218 conjugacy classes of subgroups — too many to draw. Only its 298 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);(4,6)(7,8,11,15,13,16,12,21)(9,17,18,22,10,19,20,14)(23,24)(25,31,33,38)(26,28)(27,30)(29,32,36,35)(34,37);(3,4,5,6)(7,9,13,10)(8,14,16,22)(11,20,12,18)(15,19,21,17)(24,28)(27,34)(29,36)(32,35);(7,10,13,9)(8,14,16,22)(11,20,12,18)(15,17,21,19)(23,25)(24,29)(26,33)(27,35)(28,36)(30,38)(31,37)(32,34)

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, 101672);