#263931

order	512
nilpotency class	3
isoclinism	isoclinic class of #263931
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18569	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

884 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=e,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh,g^b=ghi,d^c=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 171127);