#130027

order	512
nilpotency class	4
isoclinism	isoclinic class of #129769
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	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

575 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37223);