#149057

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:

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

Polycyclic presentation:

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

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, 56253);