#148420

order	512
nilpotency class	4
isoclinism	isoclinic class of #13472
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2067
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)(3,14)(4,21,25,43)(5,27)(6,12)(7,32)(8,13)(9,17)(10,24,40,26)(11,15)(16,48,51,38)(18,45)(19,37)(20,35)(22,57,49,33)(23,58,60,39)(28,41,59,64)(29,42)(30,61)(31,46)(34,47)(36,54)(44,50,56,53)(52,62,55,63);(1,3,15,47,7,18,30,19,8,20,29,54,6,31,5,17)(2,9,27,46,12,36,42,35,13,37,61,45,32,34,11,14)(4,22,28,51,26,55,60,53,25,49,59,16,24,52,23,50)(10,38,41,57,21,56,58,62,40,48,64,33,43,44,39,63);(1,4)(2,10)(3,16)(5,28)(6,26)(7,24)(8,25)(9,33)(11,41)(12,43)(13,40)(14,44)(15,23)(17,52)(18,53)(19,55)(20,51)(21,32)(22,54)(27,39)(29,60)(30,59)(31,50)(34,63)(35,56)(36,62)(37,57)(38,45)(42,64)(46,48)(47,49)(58,61)

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=fh,h^a=hi,c^b=cf,e^b=ei,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, 55616);