#401664

order	512
nilpotency class	3
isoclinism	isoclinic class of #394199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1208 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)(18,27)(19,28)(21,35)(22,29)(23,24)(32,33)(34,36)(37,40);(1,3)(2,5)(4,7)(6,8)(9,10,16,28,14,19,13,15)(11,17,29,38,22,30,20,26)(12,18,24,32,25,33,23,27)(21,31,35,40,36,39,34,37);(3,7)(5,8)(9,11,13,20,14,22,16,29)(10,17,15,26,19,30,28,38)(12,21,23,34,25,36,24,35)(18,31,27,37,33,39,32,40);(9,12)(10,18)(11,21)(13,23)(14,25)(15,27)(16,24)(17,31)(19,33)(20,34)(22,36)(26,37)(28,32)(29,35)(30,39)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,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, 308860);