#131441

order	512
nilpotency class	4
isoclinism	isoclinic class of #131152
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

642 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,f^c=fg,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38637);