#6867477

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1460 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)(3,7)(4,6)(5,8)(17,18,21,25,22,26,34,40)(19,23,29,37,28,36,44,47)(20,30,31,38,32,39,33,24)(27,41,43,46,42,48,45,35);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,28)(18,23,26,36)(20,27,32,42)(21,29,34,44)(24,35,38,46)(25,37,40,47)(30,41,39,48)(31,43,33,45);(10,14)(15,16)(17,20,22,32)(18,24,26,38)(19,27,28,42)(21,33,34,31)(23,35,36,46)(25,39,40,30)(29,45,44,43)(37,48,47,41);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(19,28)(23,36)(27,42)(29,44)(35,46)(37,47)(41,48)(43,45);(3,8)(5,7)(9,12,13,11)(10,15,14,16)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=h,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6774673);