#349414

order	512
nilpotency class	3
isoclinism	isoclinic class of #346882
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×H(3,𝔽₄)	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 973 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=ei,b^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,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, 256610);