#503748

order	512
nilpotency class	4
isoclinism	isoclinic class of #503748
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#26044
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

482 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=fh,d^2=e,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=ghi,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, 410944);