#489723

order	512
nilpotency class	4
isoclinism	isoclinic class of #489719
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61089	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

958 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)(3,6)(4,7)(5,8)(9,10,14,17)(11,16,20,32)(12,15,25,28)(13,26,24,19)(18,33,29,23)(21,27,34,39)(22,37,35,31)(30,40,38,36);(2,6)(3,4)(7,8)(10,15)(12,23)(13,24)(16,27)(17,28)(18,19)(21,36)(22,35)(25,33)(26,29)(30,31)(32,39)(34,40)(37,38);(2,7)(3,4)(6,8)(9,11)(10,16)(12,21)(13,22)(14,20)(15,27)(17,32)(18,30)(19,31)(23,36)(24,35)(25,34)(26,37)(28,39)(29,38)(33,40);(2,8)(6,7)(10,17)(11,20)(15,28)(18,29)(19,26)(21,34)(22,35)(36,40)

Polycyclic presentation:

2^9|a^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fh,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396919);