#350108

order	512
nilpotency class	3
isoclinism	isoclinic class of #346918
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×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

972 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 972 conjugacy classes of subgroups — too many to draw. Only its 152 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)(9,10,14,18)(11,15,13,17)(12,21,22,23)(16,19,24,20)(25,26,30,33)(27,31,29,37)(28,35,36,39)(32,38,40,34);(2,5)(6,8)(9,11,14,13)(10,15,18,17)(12,19,22,20)(16,21,24,23)(25,27,30,29)(26,31,33,37)(28,34,36,38)(32,39,40,35);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,20)(14,22)(15,21)(17,23)(18,24)(25,28,30,36)(26,32,33,40)(27,34,29,38)(31,39,37,35);(9,12)(10,16)(11,20)(13,19)(14,22)(15,23)(17,21)(18,24)(27,29)(31,37)(34,38)(35,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ei,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, 257304);