#451179

order	512
nilpotency class	3
isoclinism	isoclinic class of #451178
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15987
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

731 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)(3,8,15,33)(4,18)(5,23)(6,27)(7,11)(9,20)(10,25)(12,29)(13,40,42,59)(14,44,24,54)(16,34,17,32)(19,49)(21,36)(22,37)(26,39)(28,38)(30,47,57,43)(31,45,58,46)(35,53)(41,56,50,63)(48,52)(51,55)(60,62,64,61);(1,3,6,16,7,15,29,17)(2,8,12,32,11,33,27,34)(4,13,20,43,22,42,21,47)(5,24,25,46,26,14,28,45)(9,30,18,40,36,57,37,59)(10,31,23,54,38,58,39,44)(19,50,51,62,53,41,52,61)(35,56,55,64,49,63,48,60);(1,4)(2,9,11,36)(3,13,15,42)(5,19)(6,20)(7,22)(8,30)(10,35,38,49)(12,18,27,37)(14,41,24,50)(16,43,17,47)(21,29)(23,55,39,48)(25,51)(26,53)(28,52)(31,56)(32,40)(33,57)(34,59)(44,60)(45,61,46,62)(54,64)(58,63);(1,5,7,26)(2,10,11,38)(3,14,15,24)(4,19,22,53)(6,28,29,25)(8,31,33,58)(9,35,36,49)(12,39,27,23)(13,41,42,50)(16,46,17,45)(18,48,37,55)(20,52,21,51)(30,56,57,63)(32,44,34,54)(40,60,59,64)(43,62,47,61)

Polycyclic presentation:

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

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, 358375);