#326102

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

610 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 138 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,22)(13,23)(14,18)(15,20)(17,19)(21,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,38)(36,40)(37,39);(1,3,5,4)(2,6,7,8)(9,11,19,12,14,21,13,20)(10,15,23,24,18,22,17,16)(25,27,29,36,28,35,37,34)(26,30,32,38,31,33,39,40);(1,4,5,3)(2,6,7,8)(9,11,13,20,14,21,19,12)(10,15,17,16,18,22,23,24)(26,31)(27,34)(29,37)(30,38)(33,40)(35,36);(2,7)(6,8)(9,12,19,21,14,20,13,11)(10,16,23,15,18,24,17,22)(27,35)(30,33)(34,36)(38,40)

Polycyclic presentation:

2^9|b^2=egi,c^2=ei,d^2=fhi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,f^a=fh,c^b=cg,d^b=dhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233298);