#219522

order	512
nilpotency class	3
isoclinism	isoclinic class of #219518
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×D₈	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

579 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:

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

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=dei,e^a=eh,c^b=cei,f^b=fh,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126718);