#474437

order	512
nilpotency class	3
isoclinism	isoclinic class of #455830
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2648
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16966
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

518 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381633);