#238877

order	512
nilpotency class	3
isoclinism	isoclinic class of #238875
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

908 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 188 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)(3,6)(4,7)(5,8)(9,10,14,18)(11,19,22,33)(12,23,25,16)(13,27,29,17)(15,28,31,21)(20,34,24,38)(26,32,40,37)(30,39,36,35);(1,2)(3,7)(4,6)(5,8)(9,11,14,22)(10,15,18,31)(12,24,25,20)(13,28,29,21)(16,30,23,36)(17,33,27,19)(26,35,40,39)(32,38,37,34);(2,6)(3,4)(7,8)(9,12,13,26,14,25,29,40)(10,16,17,32,18,23,27,37)(11,20,21,35,22,24,28,39)(15,30,19,34,31,36,33,38);(12,25)(16,23)(20,24)(26,40)(30,36)(32,37)(34,38)(35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146073);