#326061

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

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

normal subgroups only

This group has 576 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=egh,c^2=ei,d^2=f,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dhi,d^c=dh,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, 233257);