#101241

order	512
nilpotency class	3
isoclinism	isoclinic class of #100939
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

598 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=h,d^2=gh,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,e^b=ei,f^b=fh,d^c=di,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, 8437);