#331783

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

672 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 142 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)(3,7)(4,6)(5,8)(9,10,14,18)(11,15,13,16)(12,20,21,23)(17,22,24,19)(25,26,30,29)(27,31,34,37)(28,32,36,38)(33,39,40,35);(1,3)(2,5)(4,8)(6,7)(9,11,14,13)(10,15,18,16)(12,19,21,22)(17,23,24,20)(25,27,30,34)(26,31,29,37)(28,35,36,39)(32,33,38,40);(10,16)(11,13)(15,18)(17,20)(19,22)(23,24)(25,28)(26,32)(27,33)(29,38)(30,36)(31,39)(34,40)(35,37);(2,6)(3,8)(9,12,14,21)(10,17,18,24)(11,19,13,22)(15,23,16,20)(25,26,30,29)(27,31,34,37)(28,32,36,38)(33,39,40,35)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238979);