#434071

order	512
nilpotency class	3
isoclinism	isoclinic class of #419810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1193 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341267);