#504337

order	512
nilpotency class	4
isoclinism	isoclinic class of #504336
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#26529
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

460 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411533);