#148437

order	512
nilpotency class	4
isoclinism	isoclinic class of #140776
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1970
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

657 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,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,e^b=ei,d^c=dhi,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, 55633);