#131858

order	512
nilpotency class	4
isoclinism	isoclinic class of #131796
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	#61091	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

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

normal subgroups only

This group has 795 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)(9,10,15,22,17,24,16,23)(11,25,29,44,31,55,30,54)(12,32,35,48,37,66,36,52)(13,38,39,20,40,21,14,41)(18,28,45,27,47,62,46,63)(19,33,49,34,51,42,50,68)(26,56,59,71,61,70,60,43)(53,57,69,58,72,64,65,67);(5,8)(6,7)(9,11,17,31)(10,18,24,47)(12,26,37,61)(13,28,40,62)(14,27,39,63)(15,30,16,29)(19,43,51,71)(20,44,41,54)(21,25,38,55)(22,46,23,45)(32,65,66,69)(33,67,42,58)(34,64,68,57)(35,60,36,59)(48,72,52,53)(49,70,50,56);(1,3)(2,4)(5,6)(7,8)(9,12)(10,19)(11,26)(13,34)(14,42)(15,36)(16,35)(17,37)(18,43)(20,32)(21,52)(22,50)(23,49)(24,51)(25,53)(27,58)(28,64)(29,60)(30,59)(31,61)(33,39)(38,48)(40,68)(41,66)(44,65)(45,70)(46,56)(47,71)(54,69)(55,72)(57,62)(63,67)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=gi,f^2=ghi,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=eg,f^c=fi,g^c=gi,h^c=hi

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, 39054);