#341696

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
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

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

normal subgroups only

This group has 476 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ei,c^2=hi,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,f^a=fh,c^b=cg,d^b=deh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 248892);