C₂×#13684 #512892

order	512
nilpotency class	5
isoclinism	isoclinic class of #13681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13684

Subgroups

	subgroup	G/H
center, socle	C₂²	#2196
commutator, Frattini, cosocle	C₄⋊Q₈	C₂⁴
prime radical	C₂×#3293	C₂
LCS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈ ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1029 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 1029 conjugacy classes of subgroups — too many to draw. Only its 114 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);(3,4,10,9)(5,14,18,28)(6,19,21,23)(7,8,24,13)(11,22,17,16)(12,25,26,20)(15,27,31,33)(29,32,34,30);(3,5,10,18)(4,11,9,17)(6,15,21,31)(7,22,24,16)(8,14,13,28)(12,27,26,33)(19,32,23,30)(20,34,25,29);(3,6)(4,12)(5,15)(7,23)(8,25)(9,26)(10,21)(11,27)(13,20)(14,29)(16,32)(17,33)(18,31)(19,24)(22,30)(28,34)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420088);