#484107

order	512
nilpotency class	4
isoclinism	isoclinic class of #484095
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

430 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dhi,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=dh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321142, #72321228, #72321460, …
maximal normal subgroup of	#72321142, #72321228, #72321460, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391303);