#326848

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	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 971 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) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,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, 234044);