#174026

order	512
nilpotency class	3
isoclinism	isoclinic class of #173897
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

801 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 81222);