#414527

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=ef,d^2=gh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,f^c=fi,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, 321723);