#149167

order	512
nilpotency class	4
isoclinism	isoclinic class of #13545
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2101
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11316
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

413 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=dh,d^2=hi,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fh,h^b=hi,e^c=ei,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, 56363);