#130440

order	512
nilpotency class	4
isoclinism	isoclinic class of #130254
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	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

559 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 256

A minimal faithful permutation representation — 3 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=i,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dgi,e^c=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37636);