#435394

order	512
nilpotency class	3
isoclinism	isoclinic class of #435393
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15526
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

978 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:

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

Polycyclic presentation:

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

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, 342590);