#406103

order	512
nilpotency class	3
isoclinism	isoclinic class of #395587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2411
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14924
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

960 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313299);