C₂×#7399 #150175

order	512
nilpotency class	5
isoclinism	isoclinic class of #7398
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#7399

Subgroups

	subgroup	G/H
center, socle	C₂²	#1343
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#408	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 616 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) = 66
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 66 points:

(1,2);(3,4,7,12)(5,15,17,45)(6,20,23,40)(8,14,28,41)(9,13,27,42)(10,30,31,54)(11,34,37,25)(16,33,48,39)(18,35,51,63)(19,46,50,60)(21,32,53,61)(22,36,55,59)(24,44,56,66)(26,57,29,52)(38,58,64,47)(43,62,65,49);(3,5,9,19)(4,10,13,32)(6,21,25,54)(7,26,27,51)(8,29,22,18)(11,35,20,52)(12,39,42,65)(14,43,36,33)(15,44,46,38)(16,47,49,24)(17,28,50,55)(23,48,34,62)(30,58,53,56)(31,59,61,41)(37,60,40,45)(57,64,63,66)

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, 57371);