#501594

order	512
nilpotency class	4
isoclinism	isoclinic class of #501593
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24708
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3067	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

476 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408790);