#426025

order	512
nilpotency class	3
isoclinism	isoclinic class of #417630
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15345
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

966 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)(3,15,18,35,20,34,19,9)(4,21,24,42,25,10,26,38)(5,27,29,11,31,40,30,41)(16,36,45,59,46,50,22,32)(17,47,28,43,48,33,49,39)(23,53,55,37,56,61,57,51)(44,58,52,62,54,64,63,60);(1,3,8,18,7,20,6,19)(2,9,14,15,13,35,12,34)(4,22,26,16,25,45,24,46)(5,28,29,17,31,49,30,48)(10,36,42,59,21,50,38,32)(11,39,40,33,41,43,27,47)(23,54,55,52,56,44,57,63)(37,58,61,60,51,64,53,62);(1,4,8,26,7,25,6,24)(2,10,14,42,13,21,12,38)(3,16,18,45,20,46,19,22)(5,23,29,55,31,56,30,57)(9,32,15,36,35,59,34,50)(11,37,40,61,41,51,27,53)(17,44,49,63,48,54,28,52)(33,58,43,60,47,64,39,62);(1,5)(2,11)(3,17)(4,23)(6,30)(7,31)(8,29)(9,33)(10,37)(12,27)(13,41)(14,40)(15,43)(16,44)(18,49)(19,28)(20,48)(21,51)(22,52)(24,57)(25,56)(26,55)(32,58)(34,39)(35,47)(36,60)(38,53)(42,61)(45,63)(46,54)(50,62)(59,64)

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 333221);