#6574864

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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	C₂×#3285	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2079 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)(7,10)(8,9)(11,12)(13,14)(15,21)(16,22)(17,23)(18,20)(19,24)(25,36)(26,38)(27,34)(28,35)(29,39)(30,40)(31,41)(32,37)(33,42)(43,44);(6,9)(10,12)(14,20)(15,25)(16,28)(17,27)(19,32)(21,23)(26,31)(29,43)(34,36)(38,39)(40,42)(41,44);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,21)(16,29)(17,19)(18,27)(20,34)(22,39)(23,24)(25,32)(26,33)(28,31)(30,43)(35,41)(36,37)(38,42)(40,44);(13,16,18,28)(14,22,20,35)(15,26,27,43)(17,31,25,29)(19,33,32,30)(21,38,34,44)(23,41,36,39)(24,42,37,40);(13,17,18,25)(14,23,20,36)(15,19,27,32)(16,26,28,43)(21,24,34,37)(22,38,35,44)(29,33,31,30)(39,42,41,40)

Polycyclic presentation:

2^9|c^2=fi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=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, 6482060);