#6574431

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

1931 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)(15,20)(16,29)(17,22)(18,23)(19,24)(21,25)(26,41)(27,35)(28,36)(30,44)(31,37)(32,39)(33,40)(34,42)(38,43);(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,26)(17,31)(18,25)(19,28)(21,23)(22,37)(24,36)(27,32)(29,41)(30,33)(34,43)(35,39)(38,42)(40,44);(2,4)(15,25)(16,28)(18,26)(20,21)(23,41)(27,31)(29,36)(30,43)(32,34)(35,37)(38,44)(39,42);(13,16,19,25)(14,21,24,29)(15,26,28,18)(17,27,33,43)(20,23,36,41)(22,38,40,35)(30,34,31,32)(37,42,44,39);(13,17,19,33)(14,22,24,40)(15,27,28,43)(16,30,25,31)(18,34,26,32)(20,35,36,38)(21,37,29,44)(23,42,41,39)

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,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, 6481627);