#143222

order	512
nilpotency class	4
isoclinism	isoclinic class of #143219
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	#532	D₄
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

495 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,6)(4,7)(5,8)(9,10,16,22)(11,23,28,56)(12,18,32,48)(13,33,38,51)(14,21,15,20)(17,40,44,25)(19,36,52,37)(24,54,58,69)(26,49,27,55)(29,62,61,72)(30,64,35,66)(31,53,39,47)(34,42,60,43)(41,57,70,59)(45,67,71,63)(46,68,50,65);(2,6)(3,4)(7,8)(9,11,16,28)(10,17,22,44)(12,29,32,61)(13,34,38,60)(14,27,15,26)(18,45,48,71)(19,49,52,55)(20,43,21,42)(23,51,56,33)(24,39,58,31)(25,36,40,37)(30,59,35,57)(41,53,70,47)(46,54,50,69)(62,66,72,64)(63,68,67,65);(2,7)(3,4)(6,8)(9,12)(10,18)(11,24)(13,35)(14,39)(15,31)(16,32)(17,41)(19,50)(20,53)(21,47)(22,48)(23,54)(25,59)(26,61)(27,29)(28,58)(30,38)(33,66)(34,67)(36,65)(37,68)(40,57)(42,71)(43,45)(44,70)(46,52)(49,72)(51,64)(55,62)(56,69)(60,63)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,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=ehi,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50418);