#415976

order	512
nilpotency class	3
isoclinism	isoclinic class of #394321
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1589 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)(3,7)(4,5)(6,8)(9,10)(11,20)(12,25)(13,30)(14,18)(15,19)(16,22)(17,27)(21,38)(23,33)(24,32)(26,37)(28,35)(29,36)(31,40)(34,39);(2,5)(6,7)(9,11,13,22,15,24,14,23)(10,16,18,20,19,33,30,32)(12,26,27,21,29,39,28,40)(17,34,25,38,35,37,36,31);(9,12,15,29)(10,17,19,35)(11,21,24,40)(13,28,14,27)(16,31,33,38)(18,36,30,25)(20,37,32,34)(22,26,23,39);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

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

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, 323172);