#98111

order	512
nilpotency class	3
isoclinism	isoclinic class of #97300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

627 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5307);