#121646

order	512
nilpotency class	3
isoclinism	isoclinic class of #102045
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

534 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 142 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)(3,8)(4,6)(5,7)(10,14,11,15)(17,18,22,26)(19,27,30,32)(20,25,21,24)(23,29,31,28);(1,3,5,4)(2,6,7,8)(9,10,13,11)(12,15,16,14)(17,19,22,30)(18,23,26,31)(20,29,21,28)(24,32,25,27);(1,4,5,3)(2,6,7,8)(9,11,13,10)(12,14,16,15)(18,24)(19,28)(20,21)(23,31)(25,26)(29,30)

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=ei,f^c=fh,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28842);