#349457

order	512
nilpotency class	3
isoclinism	isoclinic class of #346918
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

793 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 793 conjugacy classes of subgroups — too many to draw. Only its 160 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)(9,10,13,16)(11,20,21,23)(12,22,17,24)(14,19,15,18)(26,28)(27,33)(30,37)(31,39)(34,40)(35,38);(1,3,4,8)(2,5,6,7)(10,16)(12,17)(18,19)(20,23)(25,26)(27,31)(28,29)(30,32)(33,40)(34,35)(36,37)(38,39);(9,11)(10,17)(12,16)(13,21)(14,24)(15,22)(18,23)(19,20)(25,27)(26,31)(28,34)(29,35)(30,38)(32,39)(33,37)(36,40);(9,12,13,17)(10,11,16,21)(14,23,15,20)(18,22,19,24)(25,28,29,26)(27,34,35,31)(30,32,37,36)(33,40,38,39)

Polycyclic presentation:

2^9|a^2=ei,b^2=eh,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,d^b=dh,f^c=fh,g^c=gi,f^d=fi,g^d=ghi

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, 256653);