C₂×#10303 #233321

order	512
nilpotency class	3
isoclinism	isoclinic class of #10300
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#10303

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1280 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1280 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 128

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140517);