#131665

order	512
nilpotency class	4
isoclinism	isoclinic class of #131664
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#62439	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

584 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,f^b=fgh,h^b=hi,e^c=egi,f^c=fg

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38861);