C₂×#12449 #490489

order	512
nilpotency class	4
isoclinism	isoclinic class of #12416
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#12449

Subgroups

	subgroup	G/H
center, socle	C₂³	#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

510 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 256

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

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397685);