#349601

order	512
nilpotency class	3
isoclinism	isoclinic class of #346740
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×#555	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

891 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 891 conjugacy classes of subgroups — too many to draw. Only its 176 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)(11,16)(12,21)(13,18)(14,23)(15,19)(17,22)(20,24)(25,26,30,34)(27,31,29,33)(28,36,38,39)(32,35,40,37);(1,3,4,8)(2,5,6,7)(9,11,15,12)(10,16,19,21)(13,22,14,20)(17,23,24,18)(25,27,30,29)(26,31,34,33)(28,35,38,37)(32,36,40,39);(9,12,15,11)(10,17,19,24)(13,22,14,20)(16,23,21,18)(25,28,30,38)(26,32,34,40)(27,35,29,37)(31,36,33,39);(9,13,15,14)(10,18,19,23)(11,20,12,22)(16,24,21,17)(26,33)(27,29)(31,34)(32,39)(35,37)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256797);