#104098

order	512
nilpotency class	3
isoclinism	isoclinic class of #104053
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	C₂⁴	#120
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

554 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 96 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)(3,5)(4,7)(6,8)(9,10,14,17)(11,18,20,23)(12,16,13,21)(15,22,24,19)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,39)(38,40);(3,8)(5,6)(9,11,14,20)(10,15,17,24)(12,19,13,22)(16,23,21,18)(25,27,29,34)(26,30,32,37)(28,33,35,40)(31,38,39,36);(2,5)(3,8)(6,7)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)(25,28)(26,31)(27,33)(29,35)(30,36)(32,39)(34,40)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11294);