#176297

order	512
nilpotency class	3
isoclinism	isoclinic class of #176173
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
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 802 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 83493);