#483810

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	C₂×#509	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

479 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)(9,10,12,17,14,20,27,19)(11,21,24,37,25,36,22,16)(13,28,26,18,31,38,39,32)(15,29,34,23,35,40,33,30);(9,11)(10,15)(12,22)(13,29)(14,25)(16,32)(17,33)(18,37)(19,34)(20,35)(21,38)(23,39)(24,27)(26,30)(28,36)(31,40);(1,3,4,6,5,7,8,2)(9,11)(10,16,20,37)(12,24)(13,30)(14,25)(15,32,35,18)(17,21,19,36)(22,27)(23,31)(26,29)(28,33,38,34)(39,40);(9,12,14,27)(10,17,20,19)(11,22,25,24)(13,26,31,39)(15,33,35,34)(16,36,37,21)(18,38,32,28)(23,29,30,40)

Polycyclic presentation:

2^9|a^2=f,c^2=fgi,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cgh,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	#72216726, #72216759, #72216898, …
maximal normal subgroup of	#72216726, #72216759, #72216898, …
maximal quotient of	C₂×#483810, #72318646, #72318937, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391006);