#405745

order	512
nilpotency class	3
isoclinism	isoclinic class of #394275
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1139 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,22)(13,26)(14,17)(16,24)(18,29)(19,36)(20,34)(21,31)(23,39)(25,35)(27,32)(28,33)(30,37)(38,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,23,25,19)(13,27)(14,21)(16,30,33,40)(17,31)(18,34)(20,29)(22,39,35,36)(24,37,28,38)(26,32);(9,12)(10,16)(11,19)(13,28)(14,25)(15,30)(17,33)(18,35)(20,38)(21,23)(22,26)(24,29)(27,37)(31,40)(32,39)(34,36);(1,4,5,3)(2,6,7,8)(10,17)(13,29)(15,31)(16,33)(20,27)(24,28)(30,40)(37,38)

Polycyclic presentation:

2^9|b^2=ehi,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,h^c=hi,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, 312941);