#474179

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	#16902
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

695 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=egh,c^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,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, 381375);