#328358

order	512
nilpotency class	3
isoclinism	isoclinic class of #326925
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	#2867	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

771 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 771 conjugacy classes of subgroups — too many to draw. Only its 136 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)(11,15,20,22)(12,16,13,17)(19,23,21,24)(25,26)(27,30)(28,31)(29,33)(32,37)(34,39)(35,38)(36,40);(2,5)(6,8)(9,11,14,20)(10,15,18,22)(12,21,13,19)(16,24,17,23)(25,27,29,34)(26,30,33,39)(28,36,37,35)(31,40,32,38);(3,7)(6,8)(10,16)(11,19)(12,13)(17,18)(20,21)(23,24)(26,31)(28,37)(30,38)(32,33)(35,36)(39,40);(1,3,4,7)(2,6,5,8)(11,20)(15,22)(19,21)(23,24)(27,34)(30,39)(35,36)(38,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,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, 235554);