#148362

order	512
nilpotency class	4
isoclinism	isoclinic class of #140148
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1926
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11121
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

661 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:

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

Polycyclic presentation:

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

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, 55558);