#137718

order	512
nilpotency class	3
isoclinism	isoclinic class of #9345
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

521 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 521 conjugacy classes of subgroups — too many to draw. Only its 104 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)(3,5,7,8)(4,6)(9,10,14,21,16,22,37,51,15,19,23,45,13,20,34,48)(11,17,27,40,26,39,57,67,25,41,35,49,32,46,62,69)(12,28,31,60,30,44,61,50,29,47,36,63,33,18,42,43)(24,52,54,70,56,38,64,66,55,59,58,72,53,65,71,68);(1,3,4,7)(2,5)(6,8)(9,11,23,27,16,26,34,57,15,25,14,35,13,32,37,62)(10,17,21,49,22,39,51,69,19,41,45,40,20,46,48,67)(12,24,31,58,30,56,61,71,29,55,36,54,33,53,42,64)(18,38,50,66,28,59,63,72,44,65,43,68,47,52,60,70);(9,12)(10,18)(11,24)(13,33)(14,36)(15,29)(16,30)(17,38)(19,44)(20,47)(21,50)(22,28)(23,31)(25,55)(26,56)(27,58)(32,53)(34,61)(35,54)(37,42)(39,59)(40,68)(41,65)(43,45)(46,52)(48,60)(49,66)(51,63)(57,71)(62,64)(67,70)(69,72)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=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, 44914);