#326052

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

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

normal subgroups only

This group has 554 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)(9,10,15,21)(11,22,24,18)(12,16,23,17)(13,20,14,19)(25,26)(27,30)(28,31)(29,32)(33,37)(34,40)(35,39)(36,38);(1,3,5,4)(2,6,7,8)(9,11,13,12,15,24,14,23)(10,16,19,22,21,17,20,18)(25,27,33,34,28,35,29,36)(26,30,37,40,31,39,32,38);(1,4,5,3)(2,6,7,8)(9,11,13,12,15,24,14,23)(10,17,19,18,21,16,20,22)(26,31)(27,34)(29,33)(30,38)(35,36)(39,40);(2,7)(6,8)(9,12,13,24,15,23,14,11)(10,18,19,16,21,22,20,17)(27,35)(30,39)(34,36)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=e,d^2=fhi,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, 233248);