#135194

order	512
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	#1820
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 366 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ci,f^b=fi,e^c=ei,f^c=fgi,f^e=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42390);