#134260

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	#1799
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

408 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 114 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)(12,18)(13,27)(14,19,29,34)(15,20)(16,23)(17,26,33,28)(22,32)(24,35,39,30)(25,36)(31,37,40,38);(9,11,13,23,15,25,12,22)(10,16,18,21,20,32,27,36)(14,24,28,38,29,39,26,37)(17,30,19,31,33,35,34,40);(9,12,15,13)(10,17)(11,22,25,23)(14,26,29,28)(16,30)(18,19)(20,33)(21,31)(24,37,39,38)(27,34)(32,35)(36,40)

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=g,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,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, 41456);