C₂×#13710 #512970

order	512
nilpotency class	5
isoclinism	isoclinic class of #13709
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#13710

Subgroups

	subgroup	G/H
center, socle	C₂²	#2226
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

358 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 138 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);(4,11)(5,15,18,33)(6,9)(7,20)(8,14)(12,28,17,34)(13,22)(16,29,25,31)(19,30)(21,24,32,26)(23,27);(3,4,9,8,10,14,6,11)(5,12,25,24,18,17,16,26)(7,13,22,20,23,30,19,27)(15,21,29,34,33,32,31,28);(3,5)(4,12)(6,16)(7,21)(8,24)(9,25)(10,18)(11,26)(13,29)(14,17)(15,27)(19,28)(20,33)(22,34)(23,32)(30,31)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#513022, C₂×#513025, C₂×#512972
maximal normal subgroup of	C₂×#513022, C₂×#513025, C₂×#512972

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420166);