#391717

order	512
nilpotency class	3
isoclinism	isoclinic class of #213744
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1744 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1744 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262956, #72263055, #72267103
maximal normal subgroup of	#72262956, #72263055, #72267103

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298913);