#98648

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, prime radical	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 552 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,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, 5844);