#305421

order	512
nilpotency class	3
isoclinism	isoclinic class of #244927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

640 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 154 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212617);