#130682

order	512
nilpotency class	4
isoclinism	isoclinic class of #130498
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
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 ◁ C₂×#556 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 654 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,d^2=hi,e^2=i,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=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, 37878);