#421643

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

1026 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=e,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,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, 328839);