#304738

order	512
nilpotency class	3
isoclinism	isoclinic class of #271387
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 792 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) = 256

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

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

Polycyclic presentation:

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

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, 211934);