#448303

order	512
nilpotency class	3
isoclinism	isoclinic class of #419445
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1076 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 355499);