#171871

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	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

738 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 148 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)(3,6)(7,8)(10,15)(11,16)(12,13)(17,18)(19,26)(20,24)(21,30)(22,25)(23,27)(28,32)(29,31);(1,2,4,5)(3,6,7,8)(9,10)(11,12)(13,15)(14,16)(17,19,22,29)(18,23,25,32)(20,27,21,28)(24,31,30,26);(1,3)(2,6)(4,7)(5,8)(9,11,14,15)(10,12,16,13)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,31,32,26);(17,20,22,21)(18,24,25,30)(19,28,29,27)(23,26,32,31)

Polycyclic presentation:

2^9|b^2=gi,c^2=g,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, 79067);