#129318

order	512
nilpotency class	3
isoclinism	isoclinic class of #8465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1779
prime radical	#3449	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

816 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

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

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, 36514);