#168744

order	512
nilpotency class	3
isoclinism	isoclinic class of #168497
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

741 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 741 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 75940);