#147376

order	512
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
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

690 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)(5,6,9,14,10,15,25,36,11,16,26,37,27,38,52,64)(7,12,19,30,20,31,44,57,18,29,40,54,43,56,46,59)(8,13,23,34,24,35,51,63,22,33,49,61,50,62,67,68)(17,39,42,65,21,45,48,66,41,28,53,55,47,32,58,60);(2,4)(5,7)(6,12)(8,21,22,47)(9,19)(10,20)(11,18)(13,32,33,45)(14,30)(15,31)(16,29)(17,24,41,50)(23,48,49,58)(25,44)(26,40)(27,43)(28,35,39,62)(34,60,61,66)(36,57)(37,54)(38,56)(42,51,53,67)(46,52)(55,63,65,68)(59,64);(5,8,9,23,10,24,25,51,11,22,26,49,27,50,52,67)(6,13,14,34,15,35,36,63,16,33,37,61,38,62,64,68)(7,17,40,53,20,21,46,58,18,41,19,42,43,47,44,48)(12,28,54,65,31,32,59,66,29,39,30,55,56,45,57,60)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,f^2=h,h^2=i,b^a=bd,c^a=ce,c^b=ch,e^b=ei,d^c=di,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, 54572);