#410685

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1034 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)(7,12)(8,11)(9,10)(13,14)(15,25)(16,30)(17,23)(18,21)(19,24)(20,28)(22,32)(26,34)(27,41)(29,36)(31,37)(33,38)(35,43)(39,42)(40,44);(2,4)(6,10)(11,12)(13,15,17,28,19,29,18,26)(14,20,21,25,24,34,23,36)(16,27,32,42,31,43,33,40)(22,35,30,39,38,41,37,44);(5,7,9,8)(6,11,10,12)(14,21,24,23)(15,26,29,28)(16,31)(20,34)(22,37,38,30)(25,36)(27,42,43,40)(32,33);(5,8,9,7)(6,11,10,12)(13,16,18,33,19,31,17,32)(14,22,23,37,24,38,21,30)(15,27,26,40,29,43,28,42)(20,35,36,44,34,41,25,39)

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=fgi,d^2=f,e^2=i,f^2=i,g^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,g^b=gi,d^c=di

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, 317881);