#495066

order	512
nilpotency class	4
isoclinism	isoclinic class of #33595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#459
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1947
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

643 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)(3,15,18,42,19,43,30,38)(4,20,23,10,21,37,24,35)(5,25,27,40,29,39,28,11)(9,26,33,48,34,57,32,17)(16,44,47,41,45,63,55,62)(22,36,53,56,51,61,54,50)(31,58,60,64,59,46,49,52);(1,3,6,18,8,19,7,30)(2,9,13,33,14,34,12,32)(4,16,23,47,21,45,24,55)(5,26,27,48,29,57,28,17)(10,31,37,60,35,59,20,49)(11,38,25,15,40,42,39,43)(22,52,53,58,51,64,54,46)(36,44,56,41,61,63,50,62);(1,4,6,23,8,21,7,24)(2,10,12,20,14,35,13,37)(3,16,18,47,19,45,30,55)(5,22,27,53,29,51,28,54)(9,31,32,49,34,59,33,60)(11,36,39,50,40,61,25,56)(15,41,38,44,43,62,42,63)(17,46,26,52,48,58,57,64);(4,21)(10,35)(16,45)(20,37)(22,51)(23,24)(31,59)(36,61)(41,62)(44,63)(46,58)(47,55)(49,60)(50,56)(52,64)(53,54)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,e^b=eh,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402262);