C₂×#7343 #149535

order	512
nilpotency class	5
isoclinism	isoclinic class of #7339
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#7343

Subgroups

	subgroup	G/H
center, socle	C₂²	#1316
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	#407	C₂³
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂²×C₄ ▷ 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);(3,4,7,12,8,11,22,27,9,13,21,30,6,19,20,29)(5,14,16,31,17,25,33,28,18,26,32,34,15,10,23,24);(3,5,9,18)(4,10,13,25)(6,17,8,15)(7,16,21,32)(11,26,19,14)(12,28,30,24)(20,33,22,23)(27,31,29,34)

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, 56731);