#442355

order	512
nilpotency class	3
isoclinism	isoclinic class of #442355
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15709
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

891 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 349551);