#219541

order	512
nilpotency class	3
isoclinism	isoclinic class of #219532
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

586 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=efh,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=ce,d^b=di,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, 126737);