C₂×#7441 #150308

order	512
nilpotency class	5
isoclinism	isoclinic class of #7440
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×#7441

Subgroups

	subgroup	G/H
center, socle	C₂²	#1353
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	#423	C₂³
prime radical	C₂⁴	#104
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#423 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

292 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 74 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)(5,13,15,25,17,30,16,33)(6,10,19,26)(7,21)(8,11)(9,14,23,31,24,32,29,18)(12,22)(20,28,34,27);(3,5)(4,9,11,24)(6,18,19,31)(7,16)(8,17)(10,25)(12,23,21,29)(13,28)(14,20,32,34)(15,22)(26,33)(27,30)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57504);