#442133

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

Polycyclic presentation:

2^9|a^2=efg,b^2=efg,c^2=efi,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, 349329);