#147007

order	512
nilpotency class	4
isoclinism	isoclinic class of #13291
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1993
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10843
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

525 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,d^2=h,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,c^b=ch,e^b=ei,f^b=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, 54203);