#147482

order	512
nilpotency class	4
isoclinism	isoclinic class of #13033
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1879
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10931
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

713 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314212, #72314275
maximal normal subgroup of	#72314212, #72314275

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54678);