#147388

order	512
nilpotency class	3
isoclinism	isoclinic class of #11000
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#390
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10922
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

679 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,f^2=h,h^2=i,b^a=bd,c^a=ce,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54584);