C₂×#7388 #150068

order	512
nilpotency class	5
isoclinism	isoclinic class of #1418
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7388

Subgroups

	subgroup	G/H
center	C₂×C₄	#357
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×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 250 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);(3,4,7,13,9,14,31,41,10,16,26,46,34,15,8,32)(5,17,19,49,21,51,33,50,23,52,20,58,22,59,54,40)(6,24,27,42,29,60,53,44,30,48,64,43,65,61,28,12)(11,35,36,56,38,63,37,18,39,25,45,55,47,62,66,57);(3,5)(4,11)(6,25)(7,20)(8,33)(9,21)(10,23)(12,40)(13,45)(14,38)(15,47)(16,39)(17,48)(18,53)(19,26)(22,34)(24,52)(27,56)(28,57)(29,62)(30,35)(31,54)(32,37)(36,46)(41,66)(42,49)(43,58)(44,50)(51,61)(55,64)(59,60)(63,65)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#151086, D₃×#7388, #72286801, …
maximal normal subgroup of	C₂×#151086, #72286801, #72286915, …
maximal quotient of	D₃×#7388, #72286929

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57264);