#441806

order	512
nilpotency class	3
isoclinism	isoclinic class of #428059
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

875 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)(5,6,9,16)(7,12)(8,21)(10,15,11,14)(13,24)(17,32,33,36)(18,27)(19,26)(20,31)(22,29)(23,30)(25,34,28,35);(5,7,11,19,9,18,10,20)(6,12,14,26,16,27,15,31)(8,17,23,35,22,33,24,34)(13,28,21,32,30,25,29,36);(5,8,10,23,9,22,11,24)(6,13,14,29,16,30,15,21)(7,17,19,34,18,33,20,35)(12,25,31,36,27,28,26,32);(2,4)(6,14,16,15)(7,18)(12,26,27,31)(13,29,30,21)(17,33)(19,20)(25,32,28,36)(34,35)

Polycyclic presentation:

2^9|a^2=eg,b^2=efg,c^2=efi,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,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, 349002);