#449834

order	512
nilpotency class	3
isoclinism	isoclinic class of #417700
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,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, 357030);