#134664

order	512
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1568
prime radical	C₂⁴	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

324 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6,10,18,12,19,35,46,13,17,34,16,11,38,9,33)(7,20,23,41,25,48,22,40,26,44,36,63,24,57,55,14)(8,27,29,45,31,59,28,60,32,15,37,42,30,43,61,62)(21,49,51,64,53,67,68,58,54,47,56,65,52,66,50,39);(5,7,9,22,11,24,34,23,13,26,35,55,12,25,10,36)(6,14,16,20,19,41,33,48,17,40,18,44,38,63,46,57)(8,21,28,50,30,52,29,56,32,54,61,68,31,53,37,51)(15,39,42,47,43,64,62,66,27,58,45,49,59,65,60,67);(2,4)(5,8)(6,15)(7,21)(9,28)(10,37)(11,30)(12,31)(13,32)(14,39)(16,42)(17,27)(18,45)(19,43)(20,47)(22,50)(23,56)(24,52)(25,53)(26,54)(29,34)(33,62)(35,61)(36,51)(38,59)(40,58)(41,64)(44,49)(46,60)(48,66)(55,68)(57,67)(63,65)

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,f^b=fi,f^c=fi

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, 41860);