#129773

order	512
nilpotency class	4
isoclinism	isoclinic class of #129769
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

852 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 108 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)(9,10,14,21,16,20,15,22)(11,23,26,46,28,50,27,51)(12,29,31,45,33,18,32,44)(13,34,36,19,38,48,37,47)(17,35,41,25,40,57,42,58)(24,52,54,69,56,49,55,70)(30,61,63,43,65,66,64,60)(39,62,67,72,59,71,68,53);(1,3)(2,5)(4,8)(6,7)(9,11,16,28)(10,17,20,40)(12,24,33,56)(13,35,38,57)(14,27,15,26)(18,39,29,59)(19,46,47,51)(21,42,22,41)(23,34,50,48)(25,36,58,37)(30,62,65,71)(31,55,32,54)(43,69,60,70)(44,68,45,67)(49,66,52,61)(53,64,72,63);(9,12)(10,18)(11,24)(13,30)(14,32)(15,31)(16,33)(17,39)(19,43)(20,29)(21,45)(22,44)(23,49)(25,53)(26,55)(27,54)(28,56)(34,66)(35,62)(36,64)(37,63)(38,65)(40,59)(41,68)(42,67)(46,69)(47,60)(48,61)(50,52)(51,70)(57,71)(58,72)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=dh,f^b=fghi,g^b=gi,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, 36969);