#130643

order	512
nilpotency class	4
isoclinism	isoclinic class of #130482
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂³	#379
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#557 ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

617 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 617 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gi,d^2=hi,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=ei,f^b=fghi,d^c=di,f^c=fgi,f^d=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37839);