#137987

order	512
nilpotency class	4
isoclinism	isoclinic class of #137984
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#32939	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 606 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,5,7,8)(4,6)(9,10,15,18,14,20,31,19)(11,21,17,30,24,35,23,13)(12,16,27,33,26,34,28,37)(22,39,32,25,40,38,36,29);(1,3,4,7)(2,5)(6,8)(9,11,13,10,15,23,21,19,14,24,30,20,31,17,35,18)(12,22,29,16,28,32,38,33,26,40,25,34,27,36,39,37);(9,12)(10,16)(11,22)(13,29)(14,26)(15,28)(17,36)(18,37)(19,33)(20,34)(21,38)(23,32)(24,40)(25,30)(27,31)(35,39)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=ghi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=ei,f^b=fh,d^c=di,f^c=fi

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, 45183);