#502655

order	512
nilpotency class	4
isoclinism	isoclinic class of #501244
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

768 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)(3,7)(4,6)(5,8)(9,10,16,22)(11,17,27,44)(12,28,33,62)(13,20,35,52)(14,19,37,50)(15,38,34,21)(18,45,49,30)(23,54,58,71)(24,59,36,42)(25,51,55,41)(26,53,39,43)(29,60,63,72)(31,61,65,48)(32,66,64,47)(40,67,69,56)(46,68,70,57);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,29)(13,25)(14,36)(15,39)(16,27)(18,46)(19,42)(20,51)(21,53)(22,44)(23,31)(24,37)(26,34)(28,60)(30,57)(32,67)(33,63)(35,55)(38,43)(40,47)(41,52)(45,68)(48,71)(49,70)(50,59)(54,61)(56,64)(58,65)(62,72)(66,69);(9,12,14,31,15,32,35,45,16,33,37,65,34,64,13,30)(10,18,20,47,21,48,50,62,22,49,52,66,38,61,19,28)(11,23,55,63,26,57,36,67,27,58,25,29,39,68,24,56)(17,40,59,70,43,60,51,71,44,69,42,46,53,72,41,54);(2,6)(5,7)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409851);