#414767

order	512
nilpotency class	3
isoclinism	isoclinic class of #393989
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#392
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1346 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6)(3,7,8,5)(9,10)(11,21)(12,25)(13,19)(14,18)(15,20)(16,23)(17,27)(22,30)(24,34)(26,35)(28,31)(29,33)(32,37)(36,39)(38,40);(9,11,13,23,15,24,14,28)(10,16,18,21,20,31,19,34)(12,22,27,38,26,37,29,39)(17,32,25,40,33,30,35,36);(9,12)(10,17)(11,22)(13,27)(14,29)(15,26)(16,30)(18,25)(19,35)(20,33)(21,36)(23,38)(24,37)(28,39)(31,32)(34,40);(1,3,4,8)(2,5,6,7)(12,26)(17,33)(22,37)(25,35)(27,29)(30,32)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi

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, 321963);