C₂×#7506 #151853

order	512
nilpotency class	5
isoclinism	isoclinic class of #1406
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

direct product

C₂×#7506

Subgroups

	subgroup	G/H
center	C₂×C₄	#353
commutator	C₄²	C₂×C₄²
Frattini, cosocle	#532	C₂³
socle	C₂²	#1393
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

674 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 130 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)(5,14,16,30)(7,10,21,26)(8,13,19,28)(9,18,23,17)(12,20,29,22)(15,25,31,27)(24,33,34,32);(3,5)(4,9)(6,18,19,33)(7,17)(8,15)(10,25)(11,27,28,30)(12,14)(13,24)(16,21,31,20)(22,32)(23,26,34,29)

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, 59049);