#503094

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	#61289	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

705 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=fh,d^2=ei,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, 410290);