#416060

order	512
nilpotency class	3
isoclinism	isoclinic class of #394189
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1170 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:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=e,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^c=fi

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, 323256);