#416794

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

642 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323990);