#503819

order	512
nilpotency class	4
isoclinism	isoclinic class of #503819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#26107
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

410 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)(3,13,11,6)(4,14)(5,22,24,15)(7,25,23,10)(9,20)(16,17)(18,29)(19,31)(21,27)(26,28)(30,32);(1,3,6,2,8,11,13,12)(4,16,18,28,21,31,20,30)(5,15,23,10,24,22,7,25)(9,17,27,32,29,19,14,26);(1,4,8,21)(2,9)(3,14)(5,17,24,19)(6,18,13,20)(7,26,23,32)(10,16)(11,27)(12,29)(15,30)(22,28)(25,31);(1,5,6,23,8,24,13,7)(2,10,11,22,12,25,3,15)(4,17,18,32,21,19,20,26)(9,16,27,28,29,31,14,30)

Polycyclic presentation:

2^9|a^2=fh,b^2=e,c^2=ef,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,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,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, 411015);