#496391

order	512
nilpotency class	3
isoclinism	isoclinic class of #382477
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2310
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21822
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

659 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=ceh,f^b=fi,g^c=gi,g^d=gi

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, 403587);