#502659

order	512
nilpotency class	4
isoclinism	isoclinic class of #501248
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

763 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)(12,28,30,61)(13,20,35,52)(14,19,37,50)(15,38,34,21)(18,45,47,31)(23,54,56,71)(24,59,36,42)(25,51,55,41)(26,53,39,43)(29,60,63,72)(32,62,64,49)(33,66,65,48)(40,67,69,57)(46,68,70,58);(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)(23,32)(24,37)(26,34)(28,60)(30,63)(31,58)(33,67)(35,55)(38,43)(40,48)(41,52)(45,68)(47,70)(49,71)(50,59)(54,62)(56,64)(57,65)(61,72)(66,69);(9,12,14,32,15,33,35,45,16,30,37,64,34,65,13,31)(10,18,20,48,21,49,50,61,22,47,52,66,38,62,19,28)(11,23,55,63,26,58,36,67,27,56,25,29,39,68,24,57)(17,40,59,70,43,60,51,71,44,69,42,46,53,72,41,54);(2,6)(5,7)(12,30)(18,47)(23,56)(28,61)(29,63)(31,45)(32,64)(33,65)(40,69)(46,70)(48,66)(49,62)(54,71)(57,67)(58,68)(60,72)

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,d^c=di

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, 409855);