#478279

order	512
nilpotency class	3
isoclinism	isoclinic class of #476293
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

940 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=i,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,f^d=fi,g^d=gi,h^d=hi

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, 385475);