#399220

order	512
nilpotency class	3
isoclinism	isoclinic class of #394044
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1413 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:

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

Polycyclic presentation:

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

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, 306416);