#144040

order	512
nilpotency class	4
isoclinism	isoclinic class of #143959
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#459
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

383 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=gi,e^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,c^b=cf,d^b=dg,f^b=fg,g^b=gi,d^c=dg,e^c=eh,f^c=fg,g^c=gi

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, 51236);