#412626

order	512
nilpotency class	3
isoclinism	isoclinic class of #401790
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2485
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15132
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

851 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=h,e^2=i,f^2=i,g^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,f^b=fi,g^b=gi,f^c=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, 319822);