#483807

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

480 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)(9,10,13,19,15,20,14,31)(11,21,24,17,22,35,25,33)(12,26,27,37,30,16,29,18)(23,34,39,32,28,36,40,38);(10,16)(12,27)(13,14)(17,32)(18,31)(19,37)(20,26)(21,36)(23,39)(24,25)(28,40)(29,30)(33,38)(34,35);(1,3,4,6,5,7,8,2)(9,11)(10,17,20,33)(12,28)(13,24)(14,25)(15,22)(16,32,26,38)(18,36,37,34)(19,35,31,21)(23,30)(27,40)(29,39);(11,22)(17,33)(21,35)(23,28)(24,25)(32,38)(34,36)(39,40)

Polycyclic presentation:

2^9|a^2=f,c^2=fg,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	#72216740, #72216752, #72216898, …
maximal normal subgroup of	#72216740, #72216752, #72216898, …
maximal quotient of	C₂×#483807, #72318651, #72318735, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391003);