#420393

order	512
nilpotency class	3
isoclinism	isoclinic class of #419794
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1349 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)(5,6,11,16,12,17,10,15)(7,18,9,13,21,28,20,29)(8,22,24,31,26,32,25,14)(19,33,34,36,35,27,23,30);(5,7,12,21)(6,13,17,29)(8,23,26,34)(9,11,20,10)(14,30,31,36)(15,28,16,18)(19,24,35,25)(22,27,32,33);(5,8,10,24,12,26,11,25)(6,14,15,22,17,31,16,32)(7,19,20,34,21,35,9,23)(13,27,18,30,29,33,28,36);(2,4)(5,9,10,7,12,20,11,21)(6,13,15,18,17,29,16,28)(8,23,24,19,26,34,25,35)(14,27,22,30,31,33,32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327589);