C₂×#12436 #490481

order	512
nilpotency class	4
isoclinism	isoclinic class of #12436
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12436

Subgroups

	subgroup	G/H
center, socle	C₂²	#1667
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

550 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 146 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);(4,9)(5,14)(6,10)(7,19)(12,30)(13,24)(15,16)(17,31)(18,32)(20,25)(21,26)(22,23)(27,28)(33,34);(3,4,10,23,7,12,20,24,8,13,25,30,19,22,6,9)(5,11,14,26,16,27,31,34,18,29,32,33,17,28,15,21);(3,5)(4,11)(6,15)(7,16)(8,18)(9,21)(10,14)(12,27)(13,29)(17,19)(20,31)(22,28)(23,26)(24,34)(25,32)(30,33)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216405, C₂×#492532, #72216414, …
maximal normal subgroup of	#72216405, C₂×#492532, #72216414, …
maximal quotient of	C₂×#123363

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397677);