#466603

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

943 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=fi,d^2=fi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373799);