#464427

order	512
nilpotency class	3
isoclinism	isoclinic class of #455155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
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

1043 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371623);