C₂×#7337 #149525

order	512
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7337

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×M₃₂	C₂³
socle	C₂²	#1314
prime radical	C₂²×#41	C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 372 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);(3,4,7,13,8,15,27,40,9,12,26,36,29,37,53,60,10,14,28,39,30,41,54,62,6,21,23,38,24,48,50,61)(5,16,18,42,19,43,46,63,17,11,31,32,44,34,55,57,20,35,25,51,47,59,52,65,22,33,45,56,49,58,64,66);(3,5)(4,11)(6,22)(7,25)(8,19)(9,17)(10,20)(12,35)(13,32)(14,33)(15,34)(16,21)(18,28)(23,31)(24,49)(26,45)(27,52)(29,44)(30,47)(36,51)(37,59)(38,42)(39,56)(40,57)(41,58)(43,48)(46,54)(50,55)(53,64)(60,65)(61,63)(62,66)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56721);