#401663

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,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, 308859);