#98728

order	512
nilpotency class	3
isoclinism	isoclinic class of #97444
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#35660	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

1409 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 1409 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5924);