#446203

order	512
nilpotency class	3
isoclinism	isoclinic class of #429607
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2595
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15834
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

684 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)(3,13,17,24)(4,18,22,10)(5,9)(6,11)(7,12)(8,16,25,14)(15,29)(19,23)(20,26,21,28)(27,31)(30,32);(1,3,7,17)(2,8)(4,19)(5,16,6,14)(9,13)(10,27,18,29)(11,24)(12,25)(15,21)(20,31)(22,30)(23,28,32,26);(2,9,12,11)(3,14,17,16)(10,26,18,28)(15,19,31,30);(1,4,5,21,7,22,6,20)(2,10,11,28,12,18,9,26)(3,15,16,30,17,31,14,19)(8,23,24,27,25,32,13,29)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=fg,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^d=ei,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, 353399);