#312145

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
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

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

normal subgroups only

This group has 590 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 219341);