#219579

order	512
nilpotency class	3
isoclinism	isoclinic class of #219576
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

593 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=ef,d^2=e,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=dh,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, 126775);