#507578

order	512
nilpotency class	4
isoclinism	isoclinic class of #500596
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

501 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 144 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 — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=ei,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fh,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, 414774);