#147204

order	512
nilpotency class	4
isoclinism	isoclinic class of #13175
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1943
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10881
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

487 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54400);