#6694454

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1336 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6601650);