#98478

order	512
nilpotency class	3
isoclinism	isoclinic class of #97316
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	C₄³	C₂³
prime radical	C₂×#562	C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

655 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 655 conjugacy classes of subgroups — too many to draw. Only its 92 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)(10,13)(11,15)(14,16)(19,25,26,28)(22,30,31,32);(2,5)(3,7)(9,10,12,14)(11,16,15,13)(17,18,21,24)(19,22,26,31)(20,27,29,23)(25,32,28,30);(11,15)(13,16)(17,19)(18,22)(20,28)(21,26)(23,32)(24,31)(25,29)(27,30)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=h,e^2=hi,f^2=gh,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=ehi,f^c=fhi

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, 5674);