#398393

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1586 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)(11,23,25,17)(12,28,30,36)(13,32,33,40)(15,22,16,21)(18,31,39,24)(19,29,35,34)(26,37,27,38);(5,6,8,7)(9,11,14,25)(10,17,20,23)(12,29,30,34)(13,24,33,31)(15,27,16,26)(18,32,39,40)(19,36,35,28)(21,38,22,37);(1,2,3,4)(9,12,16,31,14,30,15,24)(10,18,22,36,20,39,21,28)(11,13,27,29,25,33,26,34)(17,35,38,32,23,19,37,40);(5,7,8,6)(9,13)(10,19)(11,24)(12,27)(14,33)(15,34)(16,29)(17,36)(18,37)(20,35)(21,32)(22,40)(23,28)(25,31)(26,30)(38,39)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fi,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, 305589);