#131019

order	512
nilpotency class	4
isoclinism	isoclinic class of #130512
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2349 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

509 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 509 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=ghi,d^2=h,e^2=i,f^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=fgh,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, 38215);