#294905

order	512
nilpotency class	3
isoclinism	isoclinic class of #271643
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

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

normal subgroups only

This group has 779 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)(4,5)(6,8)(11,17)(12,19)(14,20)(16,22)(18,23)(21,24)(25,26,28,31)(27,30,35,39)(29,37,36,32)(33,38,34,40);(2,5)(6,7)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24)(25,27)(26,30)(28,35)(29,33)(31,39)(32,40)(34,36)(37,38);(9,11,12,17,13,18,19,23)(10,14,16,20,15,21,22,24)(26,31)(27,33,35,34)(30,38,39,40)(32,37);(1,3,4,8)(2,6,5,7)(10,15)(14,21)(16,22)(20,24)

Polycyclic presentation:

2^9|a^2=i,c^2=fi,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,d^b=dh,f^b=fi,g^b=gi

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, 202101);