#146586

order	512
nilpotency class	4
isoclinism	isoclinic class of #146581
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10735
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1585 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341686, #72341693, #72341708
maximal normal subgroup of	#72341686, #72341693, #72341708

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53782);