#405888

order	512
nilpotency class	3
isoclinism	isoclinic class of #395586
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2411
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14920
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

975 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)(3,13,17,36)(4,9)(5,22,24,10)(6,11)(7,12)(8,16,29,43)(14,37,18,34)(15,40)(19,30,48,50)(20,31)(21,32)(23,51,52,33)(25,26)(27,46,55,39)(28,41)(35,42)(38,58)(44,53)(45,57)(47,56,61,64)(49,60)(54,59)(62,63);(1,3,6,16,7,17,25,43)(2,8,26,36,12,29,11,13)(4,18,44,39,21,14,20,46)(5,15,23,41,24,42,52,60)(9,27,31,34,32,55,53,37)(10,28,51,40,22,49,33,35)(19,45,61,59,48,38,47,62)(30,54,56,57,50,63,64,58);(1,4,6,20,7,21,25,44)(2,9,11,31,12,32,26,53)(3,14,16,39,17,18,43,46)(5,19,23,47,24,48,52,61)(8,27,13,34,29,55,36,37)(10,30,33,56,22,50,51,64)(15,38,41,59,42,45,60,62)(28,54,35,57,49,63,40,58);(1,5)(2,10)(3,15)(4,19)(6,23)(7,24)(8,28)(9,30)(11,33)(12,22)(13,35)(14,38)(16,41)(17,42)(18,45)(20,47)(21,48)(25,52)(26,51)(27,54)(29,49)(31,56)(32,50)(34,57)(36,40)(37,58)(39,59)(43,60)(44,61)(46,62)(53,64)(55,63)

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=e,e^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,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, 313084);