C₂×#12438 #490483

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

Aliases

direct product

C₂×#12438

Subgroups

	subgroup	G/H
center, socle	C₂²	#1667
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#2213	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

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

normal subgroups only

This group has 524 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)(6,19)(7,20)(10,21)(12,26)(13,25)(15,16)(17,32)(18,27)(22,28)(23,24)(29,30)(33,34);(3,4,10,26,7,12,6,9,8,13,19,24,20,23,21,25)(5,11,27,33,16,29,15,22,18,31,14,28,17,30,32,34);(3,5)(4,11)(6,15)(7,16)(8,18)(9,22)(10,27)(12,29)(13,31)(14,19)(17,20)(21,32)(23,30)(24,28)(25,34)(26,33)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397679);