#441860

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1013 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(2,3)(5,6,9,11)(7,10,8,12)(13,14)(15,21)(16,29)(17,35)(18,25)(19,24)(20,26)(22,32)(23,36)(27,38)(28,41)(30,37)(31,44)(33,40)(34,39)(42,43);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,20,17)(14,21,26,35)(16,30,34,43)(18,36,19,28)(22,38,40,44)(23,24,41,25)(27,33,31,32)(29,37,39,42);(5,8,9,7)(6,10,11,12)(13,16,18,32,20,34,19,33)(14,22,24,29,26,40,25,39)(15,27,28,30,17,31,36,43)(21,37,23,38,35,42,41,44);(1,3,4,2)(13,17,20,15)(14,23,26,41)(16,31,34,27)(18,36,19,28)(21,24,35,25)(22,38,40,44)(29,42,39,37)(30,32,43,33)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 349056);