#424643

order	512
nilpotency class	3
isoclinism	isoclinic class of #417622
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1227 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)(11,22)(12,17)(13,20)(14,19)(15,21)(16,25)(18,29)(23,36)(24,33)(26,34)(27,39)(28,37)(30,38)(31,35)(32,40);(9,11)(10,16)(12,27)(13,31)(14,25)(15,26)(17,36)(18,32)(19,34)(20,22)(21,35)(23,29)(24,28)(30,40)(33,38)(37,39);(1,3,4,8)(2,5,6,7)(9,12)(10,17)(11,23)(13,28)(14,29)(15,30)(16,32)(18,19)(20,37)(21,38)(22,36)(24,26)(25,40)(27,31)(33,34)(35,39);(9,12)(10,18)(11,24)(13,29)(14,28)(15,30)(16,33)(17,19)(20,38)(21,37)(22,39)(23,26)(25,40)(27,31)(32,34)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331839);