#465638

order	512
nilpotency class	3
isoclinism	isoclinic class of #457092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1149 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:

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

Polycyclic presentation:

2^9|b^2=gi,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,h^c=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, 372834);