C₂×#7346 #149545

order	512
nilpotency class	5
isoclinism	isoclinic class of #1402
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#7346

Subgroups

	subgroup	G/H
center	C₂×C₄	AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂²×C₈
Frattini, cosocle	#408	C₂³
socle	C₂²	#1316
prime radical	C₂×#585	C₄
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 998 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) = 66
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56741);