#148338

order	512
nilpotency class	4
isoclinism	isoclinic class of #13389
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2033
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11096
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

653 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)(4,21,25,10)(5,26)(6,14)(7,13)(8,12)(9,18)(11,16)(17,41,48,63)(19,43)(20,51)(22,54,49,62)(23,56,46,57)(24,36,30,40)(27,37)(28,38)(29,42)(31,60,61,47)(32,58)(33,50)(34,55,52,64)(35,45,53,59)(39,44);(1,3,16,44,7,20,28,50,8,19,27,58,6,29,5,18)(2,9,26,42,14,32,37,43,12,33,38,51,13,39,11,15)(4,22,45,47,24,17,46,64,25,49,59,60,30,48,23,55)(10,34,57,41,36,31,53,54,21,52,56,63,40,61,35,62);(1,4)(2,10)(3,17)(5,23)(6,30)(7,24)(8,25)(9,31)(11,35)(12,21)(13,40)(14,36)(15,41)(16,45)(18,47)(19,48)(20,49)(22,29)(26,57)(27,59)(28,46)(32,52)(33,61)(34,39)(37,53)(38,56)(42,54)(43,63)(44,64)(50,60)(51,62)(55,58)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,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, 55534);