#421443

order	512
nilpotency class	3
isoclinism	isoclinic class of #421443
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15291
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

935 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:

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

Polycyclic presentation:

2^9|b^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,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, 328639);