#131919

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	#61290	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

571 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 571 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

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

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, 39115);