#442131

order	512
nilpotency class	3
isoclinism	isoclinic class of #428071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

897 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)(5,6,12,16,11,17,10,26)(7,18,8,13,20,30,22,14)(9,23,27,32,25,36,24,15)(19,29,21,31,33,28,34,35);(5,7,12,8,11,20,10,22)(6,13,16,30,17,14,26,18)(9,21,27,33,25,34,24,19)(15,31,23,28,32,35,36,29);(5,8,10,20,11,22,12,7)(6,14,26,18,17,13,16,30)(9,21,24,33,25,34,27,19)(15,29,36,31,32,28,23,35);(2,4)(5,9,10,24,11,25,12,27)(6,15,16,23,17,32,26,36)(7,19,8,21,20,33,22,34)(13,28,18,31,14,29,30,35)

Polycyclic presentation:

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

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, 349327);