#416824

order	512
nilpotency class	3
isoclinism	isoclinic class of #402003
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2494
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15233
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

813 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

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, 324020);