#448739

order	512
nilpotency class	3
isoclinism	isoclinic class of #421157
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2610
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15917
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

697 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)(4,9)(5,17)(6,13)(7,12)(8,10)(11,23)(15,25)(16,22)(18,19)(20,28)(21,29)(24,26)(27,30)(31,32);(1,3,7,16)(2,8,12,5)(4,18)(6,17,23,10)(9,25)(11,22,13,14)(15,30)(19,28)(20,31)(21,26)(24,27)(29,32);(1,4,6,20,7,21,23,30)(2,9,11,27,12,29,13,28)(3,15,10,26,16,31,17,18)(5,19,22,32,8,24,14,25);(1,5,7,8)(2,10)(3,13)(4,19,21,24)(6,22,23,14)(9,26)(11,16)(12,17)(15,28)(18,29)(20,32,30,25)(27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 355935);