#352646

order	512
nilpotency class	3
isoclinism	isoclinic class of #352019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

728 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 178 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)(3,6)(4,5)(7,8)(9,10)(11,19)(12,15)(13,22)(14,16)(17,20)(18,23)(21,24)(25,26)(27,33)(28,36)(29,31)(30,34)(32,37)(35,39)(38,40);(2,5)(6,8)(9,11,13,12,14,20,23,21)(10,15,18,19,16,24,22,17);(10,16)(15,24)(17,19)(18,22)(25,27,28,34,29,35,37,40)(26,30,32,33,31,38,36,39);(1,3,4,7)(2,6,5,8)(9,12,13,20,14,21,23,11)(10,17,18,15,16,19,22,24)(26,31)(30,38)(32,36)(33,39)

Polycyclic presentation:

2^9|b^2=gi,c^2=fh,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,d^b=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259842);