#410549

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

881 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)(8,10)(9,11)(13,14)(15,25)(16,30)(17,23)(18,21)(19,24)(20,28)(22,32)(26,34)(27,35)(29,36)(31,38)(33,37)(39,40)(41,43)(42,44);(2,4)(13,15,18,26,19,29,17,28)(14,20,23,36,24,34,21,25)(16,27,32,40,33,42,31,41)(22,35,30,43,38,44,37,39);(5,7,9,8)(6,10,11,12)(14,21,24,23)(15,26,29,28)(20,34)(22,37,38,30)(25,36)(27,40,42,41)(35,44)(39,43);(5,8,9,7)(6,10,11,12)(13,16,18,32,19,33,17,31)(14,22,23,30,24,38,21,37)(15,27,26,40,29,42,28,41)(20,35,36,43,34,44,25,39)

Polycyclic presentation:

2^9|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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317745);