#6495431

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469956
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

1359 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)(11,15)(14,16)(17,18,23,37,22,26,21,27)(19,28,36,24,30,43,31,40)(20,32,38,42,34,25,35,47)(29,45,48,41,33,44,46,39);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,26,40)(20,33,34,29)(21,36,23,31)(25,41,32,39)(27,43,37,28)(35,46,38,48)(42,44,47,45);(2,5)(6,8)(10,13)(14,16)(17,20,22,34)(18,25,26,32)(19,29,30,33)(21,35,23,38)(24,39,40,41)(27,42,37,47)(28,44,43,45)(31,46,36,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,30)(20,34)(24,40)(25,32)(28,43)(31,36)(35,38)(42,47);(9,11,12,15)(10,14,13,16)(18,26)(24,40)(25,32)(27,37)(28,43)(39,41)(42,47)(44,45)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6402627);