#130635

order	512
nilpotency class	4
isoclinism	isoclinic class of #130490
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂×#553	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#553 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 674 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gi,d^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fghi,e^c=ei,f^c=fg,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, 37831);