#148408

order	512
nilpotency class	4
isoclinism	isoclinic class of #13389
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2033
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11123
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

665 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)(3,15)(4,23,27,42)(5,29)(6,12)(7,33)(8,14)(9,18)(10,26,40,28)(11,16)(13,31)(17,50,53,37)(19,48)(20,36)(21,58)(22,44)(24,34,51,62)(25,39,60,38)(30,41,59,45)(32,43)(35,56)(46,52,64,54)(47,49)(55,61,57,63);(1,3,16,49,7,21,32,20,8,22,31,56,6,19,5,18)(2,9,29,48,12,35,13,44,14,36,43,58,33,47,11,15)(4,24,30,53,28,57,60,52,27,51,59,17,26,55,25,54)(10,37,41,34,23,46,39,61,40,50,45,62,42,64,38,63);(1,4)(2,10)(3,17)(5,30)(6,28)(7,26)(8,27)(9,34)(11,41)(12,42)(13,45)(14,40)(15,46)(16,25)(18,55)(19,54)(20,57)(21,52)(22,53)(23,33)(24,56)(29,38)(31,60)(32,59)(35,63)(36,62)(37,48)(39,43)(44,64)(47,61)(49,51)(50,58)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,f^b=fi,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, 55604);