#326849

order	512
nilpotency class	3
isoclinism	isoclinic class of #326835
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

833 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 833 conjugacy classes of subgroups — too many to draw. Only its 140 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)(11,13)(12,20)(14,19)(15,17)(16,23)(18,24)(21,22)(25,26,30,34)(27,31,28,36)(29,38,39,33)(32,35,40,37);(2,5)(6,8)(9,11,13,10)(12,21,22,20)(14,19,15,17)(16,24,18,23)(25,27,30,28)(26,31,34,36)(29,37,39,35)(32,33,40,38);(1,3)(2,6)(4,7)(5,8)(9,12,13,22)(10,16,11,18)(14,24,15,23)(17,21,19,20)(25,28,30,27)(26,32,34,40)(29,35,39,37)(31,38,36,33);(3,7)(6,8)(12,22)(16,18)(20,21)(23,24)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 234045);