#219599

order	512
nilpotency class	3
isoclinism	isoclinic class of #219590
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₈

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	#19701	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

625 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=ef,d^2=ei,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,d^c=di,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126795);