#404783

order	512
nilpotency class	3
isoclinism	isoclinic class of #396612
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2850
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14902
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

817 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)(3,13,16,9)(4,19)(5,7,12,8)(10,22)(14,30)(15,17,25,18)(20,24)(21,28)(23,27)(26,32)(29,31);(1,3,6,16)(2,9,11,13)(4,20)(5,15,12,25)(7,18,8,17)(10,26)(14,21)(19,24)(22,32)(23,31)(27,29)(28,30);(1,4,8,21,6,23,7,22)(2,10,5,19,11,28,12,27)(3,14,18,20,16,32,17,31)(9,24,25,26,13,29,15,30);(1,5,7,2,6,12,8,11)(3,15,18,9,16,25,17,13)(4,19,22,10,23,27,21,28)(14,30,20,24,32,26,31,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311979);