#6555770

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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

1541 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)(17,18,21,27,23,28,22,26)(19,29,33,24,31,40,32,39)(20,34,35,42,37,41,36,25)(30,44,47,48,45,38,46,43);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,31)(18,24,28,39)(20,30,37,45)(21,32,22,33)(25,38,42,44)(26,40,27,29)(34,48,41,43)(35,46,36,47);(10,13)(14,16)(17,20,23,37)(18,25,28,42)(19,30,31,45)(21,36,22,35)(24,38,39,44)(26,34,27,41)(29,43,40,48)(32,47,33,46);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(19,31)(24,39)(29,40)(30,45)(32,33)(38,44)(43,48)(46,47);(11,15)(14,16)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,35,37,36)(24,40,39,29)(25,41,42,34)(30,46,45,47)(38,43,44,48)

Polycyclic presentation:

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

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, 6462966);