#174091

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	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1043 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1043 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=i,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, 81287);