#129656

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1779
prime radical	#3529	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

577 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 577 conjugacy classes of subgroups — too many to draw. Only its 112 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(2,5)(3,7)(6,8)(9,10,14,20,15,19,34,45,16,21,35,46,13,30,33,57)(11,17,24,38,23,39,52,64,25,40,53,65,31,37,58,63)(12,26,29,55,28,18,41,43,27,44,56,68,32,42,59,67)(22,47,50,69,51,61,66,72,48,54,70,71,49,36,60,62);(1,2)(3,8)(4,6)(5,7)(9,11)(10,17)(12,22)(13,31)(14,24)(15,23)(16,25)(18,36)(19,39)(20,38)(21,40)(26,54)(27,48)(28,51)(29,50)(30,37)(32,49)(33,58)(34,52)(35,53)(41,66)(42,61)(43,62)(44,47)(45,64)(46,65)(55,71)(56,70)(57,63)(59,60)(67,72)(68,69);(3,7)(5,8)(9,12)(10,18)(11,22)(13,32)(14,29)(15,28)(16,27)(17,36)(19,44)(20,43)(21,42)(23,51)(24,50)(25,48)(26,30)(31,49)(33,59)(34,41)(35,56)(37,54)(38,62)(39,47)(40,61)(45,68)(46,67)(52,66)(53,70)(55,57)(58,60)(63,71)(64,69)(65,72)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36852);