#449972

order	512
nilpotency class	3
isoclinism	isoclinic class of #449972
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15954
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

1023 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,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, 357168);