#397447

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

1258 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=i,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,g^c=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, 304643);