#131874

order	512
nilpotency class	4
isoclinism	isoclinic class of #131824
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	#61086	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

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

normal subgroups only

This group has 800 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)(9,10,15,21,16,23,37,22)(11,24,28,43,30,55,29,63)(12,31,34,46,36,66,35,51)(13,20,14,19,38,49,40,50)(17,41,44,61,45,27,52,26)(18,39,47,68,48,32,53,33)(25,56,59,42,60,70,64,71)(54,57,65,67,72,62,69,58);(5,8)(6,7)(9,11)(10,17)(12,25)(13,27)(14,26)(15,28)(16,30)(18,42)(19,43)(20,24)(21,44)(22,52)(23,45)(29,37)(31,65)(32,67)(33,62)(34,59)(35,64)(36,60)(38,41)(39,58)(40,61)(46,72)(47,70)(48,71)(49,55)(50,63)(51,54)(53,56)(57,68)(66,69);(1,3)(2,4)(5,6)(7,8)(9,12,16,36)(10,18,23,48)(11,25,30,60)(13,33,38,68)(14,39,40,32)(15,34,37,35)(17,42,45,71)(19,31,50,66)(20,51,49,46)(21,47,22,53)(24,54,55,72)(26,58,61,67)(27,62,41,57)(28,59,29,64)(43,65,63,69)(44,70,52,56)

Polycyclic presentation:

2^9|a^2=f,c^2=i,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=egi,f^b=fghi,g^b=gi,h^b=hi,d^c=dgi,e^c=egi,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39070);