#465210

order	512
nilpotency class	3
isoclinism	isoclinic class of #454981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2619
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16516
prime radical	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1091 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 372406);