#171868

order	512
nilpotency class	3
isoclinism	isoclinic class of #171725
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

876 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 79064);