#147059

order	512
nilpotency class	4
isoclinism	isoclinic class of #12826
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1804
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10850
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

525 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54255);