#500309

order	512
nilpotency class	4
isoclinism	isoclinic class of #500308
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24061
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

745 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)(4,17)(5,22)(6,27)(7,11)(9,20)(10,26)(12,25)(13,38)(14,41)(15,32)(16,46)(18,50)(19,48)(21,35)(23,34)(24,54)(28,49)(29,39)(30,45)(31,44)(33,52)(36,56)(37,53)(40,59)(42,58)(43,63)(47,62)(51,57)(55,61)(60,64);(1,3,7,15)(2,8,11,32)(4,18)(5,14,24,43)(6,16,25,44)(9,29)(10,30,36,60)(12,31,27,46)(13,19)(17,38)(20,51)(21,47)(22,41,54,63)(23,55)(26,45,56,64)(28,40)(33,57)(34,58)(35,59)(37,61)(39,52)(42,53)(48,50)(49,62);(1,4,6,28,7,19,25,21)(2,9,12,37,11,33,27,34)(3,13,16,47,15,18,44,40)(5,23,26,52,24,53,56,20)(8,29,31,61,32,57,46,58)(10,35,22,48,36,49,54,17)(14,42,45,51,43,55,64,39)(30,59,41,50,60,62,63,38);(5,24)(10,36)(14,43)(17,48)(20,52)(22,54)(23,53)(26,56)(30,60)(35,49)(38,50)(39,51)(41,63)(42,55)(45,64)(59,62)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407505);