#472561

order	512
nilpotency class	3
isoclinism	isoclinic class of #457680
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2665
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16825
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

751 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:

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

Polycyclic presentation:

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

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, 379757);