#131336

order	512
nilpotency class	4
isoclinism	isoclinic class of #131334
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#60768	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

774 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 774 conjugacy classes of subgroups — too many to draw. Only its 100 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)(3,4,7,8)(5,6)(9,10,12,18,13,19,28,33,15,21,29,32,14,20,26,16)(11,22,23,39,30,17,34,35,25,38,27,31,24,36,40,37);(1,3)(2,4)(5,7)(6,8)(10,16)(12,26)(13,14)(17,31)(18,20)(19,32)(21,33)(22,37)(23,40)(24,30)(27,34)(28,29)(35,38)(36,39);(1,4)(2,3)(5,8)(6,7)(9,11)(10,17)(12,27)(13,30)(14,24)(15,25)(16,31)(18,37)(19,38)(20,22)(21,36)(23,29)(26,34)(28,40)(32,35)(33,39)

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dgi,e^c=ei,f^c=fgi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38532);