#130656

order	512
nilpotency class	4
isoclinism	isoclinic class of #130530
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2353 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 509 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)(12,24)(13,28,29,17)(14,18)(15,21,32,22)(16,26,34,39)(20,33)(23,31)(25,30)(27,38)(35,37,40,36);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15,18,32)(12,25)(13,22)(14,23)(16,33,34,30)(17,31,28,19)(20,27)(21,29)(24,35,38,40)(26,36)(37,39);(1,3,5,4)(2,6,7,8)(9,12,14,27)(10,16,18,34)(11,20,23,25)(13,26,29,39)(15,30,32,33)(17,24,28,38)(19,35,31,40)(21,36,22,37)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,f^b=fghi,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, 37852);