#6448032

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

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, 6355228);