#404662

order	512
nilpotency class	3
isoclinism	isoclinic class of #396611
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

919 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)(6,12,16,15)(7,17,20,24)(14,26,30,29)(19,31,33,34);(5,6,10,12,11,16,9,15)(7,18,17,28,20,25,24,13)(8,14,22,26,23,30,21,29)(19,32,31,36,33,35,34,27);(5,7,9,17,11,20,10,24)(6,13,12,25,16,28,15,18)(8,19,21,31,23,33,22,34)(14,27,26,35,30,36,29,32);(2,4)(5,8,9,21,11,23,10,22)(6,14,15,29,16,30,12,26)(7,19,17,31,20,33,24,34)(13,27,18,32,28,36,25,35)

Polycyclic presentation:

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

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, 311858);