#465716

order	512
nilpotency class	3
isoclinism	isoclinic class of #465716
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16527
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

858 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)(3,8)(4,17,19,31)(5,22)(6,11)(7,12)(9,14)(10,21,28,20)(13,26,18,27)(15,24)(16,25)(23,29,32,30);(1,3,6,15,7,16,5,14)(2,8,11,24,12,25,22,9)(4,13,21,30,19,18,20,29)(10,23,17,27,28,32,31,26);(1,3)(2,9)(4,18)(5,15)(6,14)(7,16)(8,22)(10,27)(11,25)(12,24)(13,19)(17,23)(20,29)(21,30)(26,28)(31,32);(1,4,5,20,7,19,6,21)(2,10,11,17,12,28,22,31)(3,13,14,29,16,18,15,30)(8,23,24,27,25,32,9,26)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 372912);