#427707

order	512
nilpotency class	3
isoclinism	isoclinic class of #418470
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2676
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15372
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

924 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=e,d^2=f,e^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^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, 334903);