#398391

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1634 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305587);