#473005

order	512
nilpotency class	3
isoclinism	isoclinic class of #456381
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2712
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16845
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

723 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:

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

Polycyclic presentation:

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

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, 380201);