#475183

order	512
nilpotency class	3
isoclinism	isoclinic class of #460311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2500
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17029
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

661 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:

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

Polycyclic presentation:

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

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, 382379);