C₂×#7341 #149532

order	512
nilpotency class	5
isoclinism	isoclinic class of #7338
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#7341

Subgroups

	subgroup	G/H
center, socle	C₂²	#1316
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	#407	C₂³
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#407 ◁ G
LECS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1028 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 1028 conjugacy classes of subgroups — too many to draw. Only its 82 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 — 3 generators on 34 points:

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

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, 56728);