#479062

order	512
nilpotency class	3
isoclinism	isoclinic class of #478348
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2754
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17183
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

639 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386258);