#449837

order	512
nilpotency class	3
isoclinism	isoclinic class of #417530
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×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1416 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)(11,20,24,34)(12,17)(13,18)(14,28)(15,19)(16,23,30,21)(22,37,38,35)(25,31)(26,33)(27,32)(29,36,40,39);(1,2,4,5)(3,7,8,6)(11,21)(13,14)(16,20)(18,28)(22,36)(23,24)(25,27)(29,35)(30,34)(31,32)(37,40)(38,39);(9,11)(10,16)(12,22)(13,21)(14,23)(15,24)(17,29)(18,34)(19,30)(20,28)(25,39)(26,38)(27,36)(31,35)(32,37)(33,40);(1,3,4,8)(2,6,5,7)(9,12,15,26)(10,17,19,33)(11,22,24,38)(13,27,14,25)(16,29,30,40)(18,32,28,31)(20,35,34,37)(21,36,23,39)

Polycyclic presentation:

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

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, 357033);