#325946

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

496 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=egh,c^2=fh,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233142);