#498368

order	512
nilpotency class	4
isoclinism	isoclinic class of #33854
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

591 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:

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

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,d^a=di,f^a=fi,d^b=de,e^b=eh,g^b=gh,h^b=hi,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225388, #72225568, #72225573
maximal normal subgroup of	#72225388, #72225568, #72225573

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405564);