#422137

order	512
nilpotency class	3
isoclinism	isoclinic class of #422137
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15302
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

1028 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=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, 329333);