#6442301

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,29,33,24)(20,25,35,42)(21,26,23,28)(30,45,47,38)(31,40,32,39)(34,48,46,41)(36,43,37,44);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,22,33)(18,24,27,29)(20,34,35,46)(21,32,23,31)(25,41,42,48)(26,40,28,39)(30,37,47,36)(38,44,45,43);(10,14)(13,16)(17,20,22,35)(18,25,27,42)(19,30,33,47)(21,36,23,37)(24,38,29,45)(26,43,28,44)(31,34,32,46)(39,41,40,48);(19,31)(20,35)(21,23)(24,39)(25,42)(26,28)(29,40)(30,46)(32,33)(34,47)(38,48)(41,45);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,26,27,28)(19,32,33,31)(20,36,35,37)(24,40,29,39)(25,43,42,44)(30,46,47,34)(38,48,45,41)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,d^c=di,h^c=hi,e^d=ei,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, 6349497);