#131806

order	512
nilpotency class	4
isoclinism	isoclinic class of #131804
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	#61501	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

571 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 571 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=g,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,h^a=hi,d^b=dh,e^b=eg,f^b=fghi,g^b=gi,h^b=hi,d^c=dg,e^c=egi,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39002);