#414529

order	512
nilpotency class	3
isoclinism	isoclinic class of #400688
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2544
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15175
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

748 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=ef,d^2=gh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,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, 321725);