#345668

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 552 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=efh,b^2=ehi,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defhi,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 252864);