C₂×#12367 #488931

order	512
nilpotency class	4
isoclinism	isoclinic class of #12366
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12367

Subgroups

	subgroup	G/H
center, socle	C₂²	#1629
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#3293	C₂
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

906 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 154 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);(5,13,15,32)(6,17)(7,20)(9,24,26,34)(10,28)(11,23)(14,19,16,31)(18,30)(21,22)(25,29,27,33);(3,4,8,12)(5,9,15,26)(6,18,22,28)(7,23,20,11)(10,17,30,21)(13,29,32,33)(14,27,16,25)(19,24,31,34);(3,5)(4,9)(6,19)(7,14)(8,15)(10,29)(11,25)(12,26)(13,21)(16,20)(17,32)(18,24)(22,31)(23,27)(28,34)(30,33)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396127);