#97859

order	512
nilpotency class	3
isoclinism	isoclinic class of #97718
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

677 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 677 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=gi,d^2=h,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=ei,f^b=fgh,d^c=di,f^c=fg

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5055);