#104241

order	512
nilpotency class	3
isoclinism	isoclinic class of #104181
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

639 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 639 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fghi,c^b=cf,d^b=dg,e^b=egi,d^c=dh,e^c=ei

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, 11437);