#6574608

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1927 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)(5,6,8,10)(7,11,12,9)(13,14)(15,25)(16,21)(17,22)(18,23)(19,24)(20,29)(26,36)(27,37)(28,39)(30,38)(31,44)(32,40)(33,41)(34,42)(35,43);(5,7,8,12)(6,9,10,11)(13,15,18,29)(14,20,23,25)(16,19,26,27)(17,31,32,30)(21,37,36,24)(22,38,40,44)(28,34,43,33)(35,42,39,41);(2,4)(15,26)(16,29)(19,27)(20,21)(24,37)(25,36)(28,31)(30,43)(33,34)(35,38)(39,44)(41,42);(13,16,18,26)(14,21,23,36)(15,27,29,19)(17,28,32,43)(20,24,25,37)(22,39,40,35)(30,34,31,33)(38,42,44,41);(13,17,18,32)(14,22,23,40)(15,28,29,43)(16,30,26,31)(19,34,27,33)(20,35,25,39)(21,38,36,44)(24,42,37,41)

Polycyclic presentation:

2^9|a^2=fi,b^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, 6481804);