#401075

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

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

1679 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)(6,7)(9,10,14,16,15,20,13,19)(11,21,22,33,25,30,24,17)(12,26,28,34,29,31,27,18)(23,32,37,39,38,40,36,35);(2,4)(5,6,8,7)(10,16)(11,22)(13,14)(18,31)(19,20)(21,30)(23,36)(24,25)(26,34)(27,28)(35,39)(37,38);(2,4)(9,11,15,25)(10,17,20,33)(12,23,29,38)(13,22,14,24)(16,30,19,21)(18,32,34,40)(26,35,31,39)(27,37,28,36);(5,7,8,6)(9,12)(10,18)(11,23)(13,28)(14,27)(15,29)(16,31)(17,32)(19,26)(20,34)(21,35)(22,36)(24,37)(25,38)(30,39)(33,40)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 308271);