#404238

order	512
nilpotency class	3
isoclinism	isoclinic class of #396101
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1175 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=e,d^2=gi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,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, 311434);