#416837

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

744 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=e,d^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,g^c=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, 324033);