#391742

order	512
nilpotency class	3
isoclinism	isoclinic class of #213744
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1088 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,d^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,h^c=hi,h^d=hi

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, 298938);