#143213

order	512
nilpotency class	4
isoclinism	isoclinic class of #143212
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28495	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

497 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,e^c=eh,f^c=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, 50409);