#137660

order	512
nilpotency class	4
isoclinism	isoclinic class of #137659
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

385 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44856);