#507970

order	512
nilpotency class	4
isoclinism	isoclinic class of #501330
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

493 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 493 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egi,c^2=i,d^2=g,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=cg,f^c=fh,h^c=hi,f^d=fi

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, 415166);