#405989

order	512
nilpotency class	3
isoclinism	isoclinic class of #395587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2411
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14922
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

988 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,h^c=hi,f^d=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, 313185);