#391741

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

1102 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1102 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)(19,34)(20,28)(21,29)(22,27)(23,31)(24,30)(25,33)(26,32)(35,40)(36,39)(37,38);(1,3)(2,5)(4,8)(6,7)(9,11,13,21,14,20,26,36)(10,15,18,29,17,28,32,39)(12,22,23,37,24,19,25,35)(16,27,31,38,30,34,33,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,24)(10,16,17,30)(11,19,20,22)(13,25,26,23)(15,27,28,34)(18,33,32,31)(21,37,36,35)(29,40,39,38);(3,8)(5,6)(11,20)(13,26)(15,28)(18,32)(19,22)(23,25)(27,34)(31,33)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^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, 298937);