#399145

order	512
nilpotency class	3
isoclinism	isoclinic class of #394055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1282 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)(3,8)(4,6)(5,7)(9,10)(11,21)(12,27)(13,19)(14,18)(15,20)(16,24)(17,29)(22,38)(23,37)(25,34)(26,33)(28,36)(30,35)(31,40)(32,39);(1,3,5,4)(2,6,7,8)(9,11,15,26)(10,16,20,34)(12,23,30,39)(13,25,14,24)(17,32,36,37)(18,33,19,21)(22,28,40,29)(27,38,35,31);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,22)(13,29)(14,28)(15,30)(16,31)(18,27)(19,35)(20,36)(21,37)(23,24)(25,39)(26,40)(32,33)(34,38);(9,12)(10,17)(11,23)(13,28)(14,29)(15,30)(16,32)(18,35)(19,27)(20,36)(21,38)(22,24)(25,40)(26,39)(31,33)(34,37)

Polycyclic presentation:

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

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, 306341);