#431483

order	512
nilpotency class	3
isoclinism	isoclinic class of #431482
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15439
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1042 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 338679);