#138013

order	512
nilpotency class	4
isoclinism	isoclinic class of #137980
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

388 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=g,d^2=ghi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,f^b=fh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45209);