#303617

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

854 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 854 conjugacy classes of subgroups — too many to draw. Only its 162 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,21)(13,14)(15,20)(16,22)(18,23)(19,24)(25,26,29,32)(27,33,36,40)(28,37,35,31)(30,38,39,34);(1,3,5,4)(2,6,8,7)(10,14)(15,23)(16,21)(17,24)(27,34)(28,35)(30,33)(31,37)(36,38)(39,40);(1,3,5,4)(2,7,8,6)(9,11,12,20,13,19,22,18)(10,15,16,17,14,23,21,24)(25,27,35,38,29,36,28,34)(26,30,37,40,32,39,31,33);(1,4,5,3)(2,7,8,6)(9,12,13,22)(10,16,14,21)(11,18,19,20)(15,24,23,17)

Polycyclic presentation:

2^9|a^2=i,b^2=ehi,c^2=efh,d^2=e,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, 210813);