#6574798

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2047 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)(15,20)(16,21)(17,19)(18,22)(23,38)(24,37)(25,33)(26,34)(27,39)(28,40)(29,41)(30,35)(31,36)(32,42)(43,44);(6,9)(10,12)(14,19)(15,23)(16,26)(18,30)(20,36)(24,27)(25,31)(28,43)(33,38)(37,40)(39,44)(41,42);(2,4)(5,7,8,11)(6,10,9,12)(13,15,17,25)(14,20,19,33)(16,27,26,28)(18,31,30,23)(21,39,34,40)(22,36,35,38)(24,32,43,29)(37,42,44,41);(13,16,17,26)(14,21,19,34)(15,24,25,43)(18,32,30,29)(20,37,33,44)(22,42,35,41)(23,28,31,27)(36,39,38,40);(13,15,17,25)(14,20,19,33)(16,28,26,27)(18,31,30,23)(21,40,34,39)(22,36,35,38)(24,29,43,32)(37,41,44,42)

Polycyclic presentation:

2^9|c^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=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, 6481994);