#140645

order	512
nilpotency class	4
isoclinism	isoclinic class of #13229
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 406 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=eg,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dg,e^b=ei,f^b=fg,g^b=gi,d^c=di,f^c=fi,f^d=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, 47841);