#483814

order	512
nilpotency class	4
isoclinism	isoclinic class of #483807
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	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

470 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cgh,d^a=di,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320804, #72320855, #72320866, …
maximal normal subgroup of	#72320804, #72320855, #72320866, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391010);