C₂×#10339 #233539

order	512
nilpotency class	3
isoclinism	isoclinic class of #10336
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#10339

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2142 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 2142 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) = 64

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

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

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, 140735);