#403116

order	512
nilpotency class	3
isoclinism	isoclinic class of #395187
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1200 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)(3,5,7,8)(9,10,15,19,14,12,16,20)(11,21,23,33,25,17,24,32)(13,27,28,26,29,34,30,18)(22,36,38,31,37,40,39,35);(1,3)(2,5)(4,7)(6,8)(9,11,16,23,14,25,15,24)(10,17,19,33,12,21,20,32)(13,22,28,39,29,37,30,38)(18,31,34,40,26,35,27,36);(3,7)(5,8)(9,12)(10,14)(11,21)(13,26)(15,19)(16,20)(17,25)(18,29)(22,35)(23,32)(24,33)(27,28)(30,34)(31,37)(36,39)(38,40);(9,13,16,28,14,29,15,30)(10,18,19,34,12,26,20,27)(11,22,23,39,25,37,24,38)(17,31,33,40,21,35,32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310312);