C₂×#10319 #233364

order	512
nilpotency class	3
isoclinism	isoclinic class of #10304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#10319

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1472 conjugacy classes of subgroups; only the 362 normal subgroups are shown.

normal subgroups only

This group has 1472 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)(8,13)(10,11)(14,17)(15,16)(18,19)(21,22)(23,24,28,32)(25,33,36,37)(26,31,27,30)(29,35,38,34);(4,6)(7,8,12,16)(9,17,20,21)(10,15,11,13)(14,18,22,19)(23,25,28,36)(24,29,32,38)(26,35,27,34)(30,37,31,33);(7,9,12,20)(8,14,16,22)(10,19,11,18)(13,17,15,21)(24,30)(26,27)(29,37)(31,32)(33,38)(34,35)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140560);