#244358

order	512
nilpotency class	3
isoclinism	isoclinic class of #233469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 806 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:

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

Polycyclic presentation:

2^9|a^2=eh,d^2=efhi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=dh,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=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, 151554);