#431789

order	512
nilpotency class	3
isoclinism	isoclinic class of #418498
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2673
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15446
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

860 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 338985);