#434426

order	512
nilpotency class	3
isoclinism	isoclinic class of #421903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

1176 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:

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

Polycyclic presentation:

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

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, 341622);