#6439468

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2308 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)(5,6,8,9)(7,10,11,12)(13,14)(15,26)(16,22)(17,23)(18,35)(19,25)(20,24)(21,30)(27,43)(28,41)(29,37)(31,42)(32,44)(33,40)(34,39)(36,38);(6,9)(10,12)(13,15)(14,21)(16,31)(17,32)(18,29)(19,30)(20,36)(22,39)(23,40)(24,37)(25,26)(27,33)(28,34)(35,38)(41,42)(43,44);(2,4)(13,16,19,28)(14,22,25,41)(15,27,30,17)(18,33,20,32)(21,23,26,43)(24,44,35,40)(29,34,36,31)(37,39,38,42);(5,7,8,11)(6,10,9,12)(13,17,19,27)(14,23,25,43)(15,28,30,16)(18,34,20,31)(21,22,26,41)(24,42,35,39)(29,33,36,32)(37,40,38,44);(13,18,19,20)(14,24,25,35)(15,29,30,36)(16,32,28,33)(17,34,27,31)(21,37,26,38)(22,40,41,44)(23,42,43,39)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,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, 6346664);