#6439674

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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2162 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=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, 6346870);