#138266

order	512
nilpotency class	4
isoclinism	isoclinic class of #138264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9147
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#32945	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

732 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=gh,e^2=i,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,d^c=di,e^c=ei,f^c=fi,f^d=fi

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, 45462);