#6439127

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2091 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)(3,4)(5,6,9,11)(7,12,8,10);(5,7,9,8)(6,10,11,12)(13,14)(15,23)(16,27)(17,30)(18,21)(19,26)(20,28)(22,33)(24,34)(25,37)(29,38)(31,35)(32,36)(39,41)(40,44)(42,43);(2,4)(5,8,9,7)(6,10,11,12)(13,15)(14,19)(16,24)(17,31)(18,25)(20,34)(21,35)(22,37)(23,30)(26,33)(27,39)(28,42)(29,40)(32,41)(36,44)(38,43);(13,16,18,29)(14,20,21,36)(15,24,25,40)(17,32,33,28)(19,34,35,44)(22,38,30,27)(23,39,37,43)(26,42,31,41);(14,21)(15,25)(20,36)(22,30)(24,40)(26,31)(27,38)(41,42)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346323);