#138297

order	512
nilpotency class	4
isoclinism	isoclinic class of #138296
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9149
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

414 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=h,d^2=gh,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,f^b=fhi,g^b=gi,e^c=ei,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, 45493);