C₂×#13692 #512926

order	512
nilpotency class	5
isoclinism	isoclinic class of #13691
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#13692

Subgroups

	subgroup	G/H
center, socle	C₂²	#2206
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
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

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

normal subgroups only

This group has 506 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);(4,8)(5,15)(6,9)(7,19)(11,25)(12,18)(13,14)(16,27)(17,32)(20,23)(21,29)(22,28)(24,31)(26,34)(30,33);(3,4,9,13,10,14,6,8)(5,11,24,30,17,26,16,23)(7,12,21,28,22,29,18,19)(15,20,27,34,32,33,31,25);(3,5)(4,11)(6,16)(7,20)(8,23)(9,24)(10,17)(12,27)(13,30)(14,26)(15,19)(18,25)(21,34)(22,33)(28,32)(29,31)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216258, C₂×#151634, #72216264, …
maximal normal subgroup of	#72216258, C₂×#151634, #72216264, …
maximal quotient of	#72225429

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420122);