#434081

order	512
nilpotency class	3
isoclinism	isoclinic class of #419792
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

1184 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)(5,6,11,15,10,16,9,24)(7,17,25,28,20,14,19,12)(8,21,22,31,23,13,26,35)(18,32,33,27,34,30,36,29);(5,7,9,19,10,20,11,25)(6,12,15,17,16,28,24,14)(8,18,22,33,23,34,26,36)(13,29,31,30,21,27,35,32);(5,8,11,26,10,23,9,22)(6,13,15,31,16,21,24,35)(7,18,19,33,20,34,25,36)(12,27,14,30,28,29,17,32);(2,4)(5,7)(6,14)(8,18)(9,25)(10,20)(11,19)(12,24)(13,30)(15,28)(16,17)(21,32)(22,36)(23,34)(26,33)(27,35)(29,31)

Polycyclic presentation:

2^9|a^2=eg,b^2=e,c^2=efi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 341277);