#327874

order	512
nilpotency class	3
isoclinism	isoclinic class of #326783
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	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

812 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 148 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) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ehi,c^2=hi,d^2=hi,f^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,d^c=de,f^c=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 235070);