#6432758

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431707
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1621 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)(11,14)(12,13)(15,16)(17,18)(19,27)(20,31)(21,25)(22,26)(23,29)(24,33)(28,38)(30,43)(32,40)(34,44)(35,42)(36,39)(37,41)(45,46)(47,48);(2,5)(6,8)(17,19,21,29)(18,23,25,27)(20,32,33,47)(22,36,35,30)(24,40,31,48)(26,43,42,39)(28,37,46,34)(38,44,45,41);(10,13)(14,16)(17,20,21,33)(18,24,25,31)(19,28,29,46)(22,37,35,34)(23,38,27,45)(26,44,42,41)(30,47,36,32)(39,48,43,40);(9,11,12,15)(10,14,13,16)(18,25)(19,29)(24,31)(26,42)(28,46)(30,36)(32,47)(41,44);(1,3,4,7)(2,6,5,8)(18,25)(23,27)(24,31)(26,42)(38,45)(39,43)(40,48)(41,44)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di,h^b=hi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339954);