#498367

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23025
prime radical	#33617	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

593 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=eh,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,e^b=eh,g^b=gh,h^b=hi,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405563);