#498754

order	512
nilpotency class	3
isoclinism	isoclinic class of #31077
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	H(3,𝔽₄)
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23350
prime radical	#575	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

382 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=f,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cefh,d^b=de,e^b=ei,f^b=fi,g^b=gi,d^c=dh,e^c=ei,f^d=fi,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, 405950);