C₂×#7062 #122294

order	512
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#7062

Subgroups

	subgroup	G/H
center	C₂²×C₄	#99
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
socle	C₂³	#324
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 406 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) = 128

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29490);