#130011

order	512
nilpotency class	4
isoclinism	isoclinic class of #124648
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1786
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

563 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 563 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

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

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, 37207);