C₂×#7356 #149577

order	512
nilpotency class	5
isoclinism	isoclinic class of #1406
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂×#7356

Subgroups

	subgroup	G/H
center	C₂×C₄	#353
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#423	C₂³
socle	C₂²	#1318
prime radical	C₂×#532	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#423 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 494 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);(3,4,6,11,7,10,22,28,8,13,20,30,23,12,21,31)(5,14,15,29,17,9,24,25,18,26,32,34,16,27,19,33);(3,5)(4,9)(6,19)(7,17)(8,18)(10,26)(11,29)(12,14)(13,27)(15,22)(16,23)(20,24)(21,32)(25,28)(30,34)(31,33)

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, 56773);