#453959

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2157 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)(3,4)(5,6)(7,8)(9,10,14,12)(11,21,24,17)(13,27,28,34)(15,20,16,19)(18,31,26,29)(22,30,33,25)(23,36,38,37)(32,40,35,39);(6,8)(9,11,14,24)(10,17,12,21)(13,23,28,38)(15,30,16,25)(18,32,26,35)(19,33,20,22)(27,37,34,36)(29,40,31,39);(2,4)(5,6)(7,8)(9,12,14,10)(11,22,24,33)(13,18,28,26)(15,19,16,20)(17,30,21,25)(23,37,38,36)(27,29,34,31)(32,39,35,40);(6,8)(9,13)(10,18)(11,23)(12,26)(14,28)(15,29)(16,31)(17,32)(19,27)(20,34)(21,35)(22,36)(24,38)(25,39)(30,40)(33,37)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,d^c=de,g^c=gi,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, 361155);