#98740

order	512
nilpotency class	3
isoclinism	isoclinic class of #97444
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

721 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 721 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 256

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

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

Polycyclic presentation:

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

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, 5936);