#6555442

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1532 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6462638);