#403800

order	512
nilpotency class	3
isoclinism	isoclinic class of #394756
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2436
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14882
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

926 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=g,d^2=e,e^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,f^b=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310996);