#147377

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1955
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

684 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6,10,16,11,17,32,46,12,18,33,47,9,15,25,40)(7,13,21,36,20,35,49,62,22,37,54,65,29,43,58,67)(8,23,28,56,27,41,59,45,26,44,31,60,30,14,38,42)(19,48,52,68,24,34,57,63,50,39,53,66,51,55,61,64);(2,4)(5,7)(6,13)(8,24,26,51)(9,29)(10,21)(11,20)(12,22)(14,39,41,48)(15,43)(16,36)(17,35)(18,37)(19,27,50,30)(23,55,44,34)(25,58)(28,57,31,61)(32,49)(33,54)(38,52,59,53)(40,67)(42,66,45,68)(46,62)(47,65)(56,64,60,63);(5,8,25,59,9,30,33,28,12,26,32,38,11,27,10,31)(6,14,40,56,15,44,47,42,18,41,46,60,17,23,16,45)(7,19,49,61,29,51,21,53,22,50,58,57,20,24,54,52)(13,34,62,68,43,48,36,64,37,55,67,66,35,39,65,63)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54573);