#401662

order	512
nilpotency class	3
isoclinism	isoclinic class of #394202
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
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

1176 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)(10,15)(11,20)(13,16)(17,30)(18,26)(19,27)(23,29)(25,32)(31,36)(33,37)(35,38)(39,40);(1,3)(2,5)(4,7)(6,8)(9,10,16,27,14,19,13,15)(11,17,28,30,20,32,22,25)(12,18,29,37,24,33,23,26)(21,31,38,39,34,40,35,36);(3,7)(5,8)(9,11,13,22,14,20,16,28)(10,17,15,25,19,32,27,30)(12,21,23,35,24,34,29,38)(18,31,26,36,33,40,37,39);(9,12)(10,18)(11,21)(13,23)(14,24)(15,26)(16,29)(17,31)(19,33)(20,34)(22,35)(25,36)(27,37)(28,38)(30,39)(32,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^c=fi,h^c=hi,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, 308858);