#351073

order	512
nilpotency class	3
isoclinism	isoclinic class of #348318
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	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

582 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 170 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)(3,6)(4,5)(7,8)(9,10,15,20)(11,17,21,24)(12,16,22,23)(13,19,14,18)(25,26,29,32)(27,33,35,40)(28,36,38,31)(30,37,39,34);(2,5)(6,8)(9,11,13,12,15,21,14,22)(10,16,18,17,20,23,19,24);(1,3,4,7)(2,6,5,8)(9,12,13,21,15,22,14,11)(10,17,18,23,20,24,19,16);(11,21)(12,22)(16,23)(17,24)(25,27,29,35)(26,30,32,39)(28,37,38,34)(31,40,36,33)

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,c^2=e,d^2=i,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=dh,d^c=dh,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 258269);