#130581

order	512
nilpotency class	4
isoclinism	isoclinic class of #130518
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂×#120	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2353 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

563 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 563 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=ghi,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fgh,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, 37777);