#138265

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

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

normal subgroups only

This group has 712 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,d^2=gh,e^2=i,f^2=gh,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, 45461);