#501466

order	512
nilpotency class	4
isoclinism	isoclinic class of #501466
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24607
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

491 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408662);