#446223

order	512
nilpotency class	3
isoclinism	isoclinic class of #446222
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15835
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

713 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=efg,d^2=efi,e^2=i,f^2=i,g^2=i,h^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,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 353419);