C₂×#7387 #150066

order	512
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7387

Subgroups

	subgroup	G/H
center	C₂×C₈	D₁₆
commutator	C₁₆	C₂²×C₈
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂²	#1331
prime radical	C₂×#508	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

254 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 98 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);(3,4,7,13,9,14,29,42,10,16,30,43,33,15,8,31)(5,17,19,45,20,46,51,65,22,47,52,66,21,53,32,38)(6,23,25,39,27,54,60,41,28,56,61,40,62,55,26,12)(11,24,34,48,36,57,63,50,37,59,64,49,44,58,35,18);(3,5)(4,11)(6,24)(7,19)(8,32)(9,20)(10,22)(12,38)(13,34)(14,36)(15,44)(16,37)(17,23)(18,26)(21,33)(25,48)(27,57)(28,59)(29,51)(30,52)(31,35)(39,45)(40,66)(41,65)(42,63)(43,64)(46,54)(47,56)(49,61)(50,60)(53,55)(58,62)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57262);