#474240

order	512
nilpotency class	3
isoclinism	isoclinic class of #455699
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2797
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16919
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)(3,14,16,28)(4,17,19,9)(5,20,22,10)(6,11)(7,13)(8,12)(15,29)(18,32)(21,27,23,26)(24,30)(25,31);(1,3,8,16)(2,9)(4,7,19,6)(5,15,22,31)(10,24)(11,14)(12,17)(13,28)(18,21,30,23)(20,32)(25,26)(27,29);(1,4,6,16,8,19,7,3)(2,9,11,14,12,17,13,28)(5,15,21,30,22,31,23,18)(10,25,26,32,20,29,27,24);(1,5,6,23,8,22,7,21)(2,10,13,26,12,20,11,27)(3,15,4,18,16,31,19,30)(9,24,28,25,17,32,14,29)

Polycyclic presentation:

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

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, 381436);