#326086

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

572 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=e,d^2=f,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, 233282);