#147108

order	512
nilpotency class	4
isoclinism	isoclinic class of #129125
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1776
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10863
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

495 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)(3,15)(4,22,26,56)(5,29)(6,11)(7,34)(8,14)(9,17)(10,25,42,61)(12,31)(13,30)(16,49,52,35)(18,48)(19,39)(20,36)(21,37)(23,40,57,64)(24,58,27,44)(28,43,60,41)(32,38)(33,45)(46,53)(47,51,55,59)(50,62,54,63);(1,3,5,19,30,53,33,18,8,21,31,17,7,20,6,32)(2,9,11,37,34,48,12,46,14,39,45,15,13,38,29,36)(4,23,27,51,60,16,25,50,26,57,24,59,28,52,61,54)(10,40,43,63,58,35,56,47,42,64,41,62,44,49,22,55);(1,4,6,25,7,28,31,27,8,26,33,61,30,60,5,24)(2,10,12,22,13,44,11,41,14,42,29,56,34,58,45,43)(3,16,18,51,20,23,19,54,21,52,32,59,53,57,17,50)(9,35,36,62,38,40,15,47,39,49,46,63,48,64,37,55)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,d^2=hi,e^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ei,f^c=fi,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, 54304);