#219506

order	512
nilpotency class	3
isoclinism	isoclinic class of #219498
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

583 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=ehi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cei,f^b=fh,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126702);