#149046

order	512
nilpotency class	4
isoclinism	isoclinic class of #13374
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2023
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11292
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

451 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=dh,f^b=fh,g^b=ghi,h^b=hi,f^c=fi,g^c=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, 56242);