#98640

order	512
nilpotency class	3
isoclinism	isoclinic class of #98566
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

656 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 96 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=eg,f^b=fgh,d^c=dg,e^c=eg,f^c=fgi

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, 5836);