#326938

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	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 851 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)(9,10,13,17)(11,19,21,15)(12,16,14,18)(20,24,22,23)(25,26)(27,35)(28,33)(29,34)(30,32)(31,37)(36,39)(38,40);(2,5)(6,8)(9,11,13,21)(10,15,17,19)(12,22,14,20)(16,24,18,23)(25,27,29,37)(26,31,34,35)(28,38,30,36)(32,40,33,39);(1,3)(2,6)(4,7)(5,8)(10,16)(11,20)(12,14)(17,18)(21,22)(23,24)(26,32)(28,30)(31,39)(33,34)(35,40)(36,38);(3,7)(6,8)(9,12,13,14)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,28,29,30)(26,33,34,32)(27,36,37,38)(31,40,35,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=h,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, 234134);