#148704

order	512
nilpotency class	4
isoclinism	isoclinic class of #137786
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1899
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂	#11203
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

627 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55900);