#410852

order	512
nilpotency class	3
isoclinism	isoclinic class of #403352
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1409 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)(3,5)(4,6)(7,8)(9,10)(11,19)(12,23)(13,17)(14,18)(15,22)(16,25)(20,29)(21,32)(24,36)(26,31)(27,34)(28,35)(30,37)(33,39)(38,40);(1,3,4,7)(2,5,6,8)(9,11)(10,15,18,19)(12,20)(13,21)(14,22)(16,29,23,27)(17,28,31,32)(24,33)(25,34)(26,35)(30,39,36,38)(37,40);(9,12)(10,16,18,23)(11,20)(13,24)(14,25)(15,27,19,29)(17,30,31,36)(21,33)(22,34)(26,37)(28,38,32,39)(35,40);(2,6)(5,8)(9,13,14,26)(10,17,18,31)(11,21,22,35)(12,24,25,37)(15,28,19,32)(16,30,23,36)(20,33,34,40)(27,38,29,39)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=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, 318048);