#472845

order	512
nilpotency class	3
isoclinism	isoclinic class of #455699
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2797
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16838
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

802 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380041);