#131859

order	512
nilpotency class	4
isoclinism	isoclinic class of #131804
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	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#61088	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

814 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=g,f^2=gh,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, 39055);