#146932

order	512
nilpotency class	4
isoclinism	isoclinic class of #12833
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1807
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10820
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

485 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)(3,15,19,42)(4,21)(5,12)(6,29)(7,31)(8,14)(9,17,34,48)(10,23)(11,27)(13,30)(16,32,45,58)(18,50,49,35)(20,52,51,33)(22,38)(24,39)(25,57)(26,37)(28,40)(36,56)(41,46,61,44)(43,59,62,53)(47,64,63,60)(54,55);(1,3,6,18,30,51,28,48,8,19,27,49,7,20,5,17)(2,9,12,33,31,50,11,15,14,34,40,52,13,35,29,42)(4,16,24,46,56,63,55,62,26,45,23,44,25,47,22,43)(10,32,37,59,54,64,36,41,39,58,21,53,38,60,57,61);(1,4,5,22,7,25,27,23,8,26,28,55,30,56,6,24)(2,10,11,36,13,38,12,37,14,39,29,57,31,54,40,21)(3,16,17,43,20,47,49,44,19,45,48,62,51,63,18,46)(9,32,15,41,35,60,33,59,34,58,42,61,50,64,52,53)

Polycyclic presentation:

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

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, 54128);