#147308

order	512
nilpotency class	4
isoclinism	isoclinic class of #12952
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1864
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10908
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

509 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=dhi,e^b=ehi,f^b=fi,h^b=hi,d^c=dhi,e^c=eh,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314225, #72314275, #72314278
maximal normal subgroup of	#72314225, #72314275, #72314278

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54504);