#449631

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1311 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=i,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^b=gi,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, 356827);