#445019

order	512
nilpotency class	3
isoclinism	isoclinic class of #445018
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15797
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

732 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:

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

Polycyclic presentation:

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

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, 352215);