#6560126

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1600 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)(19,23)(20,32)(21,25)(22,27)(24,34)(26,33)(28,42)(29,37)(30,38)(31,39)(35,40)(36,44)(41,46)(43,48)(45,47);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,28)(21,30)(22,31)(24,36)(25,38)(26,37)(27,39)(29,33)(32,42)(34,44)(35,45)(40,47)(41,48)(43,46);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20,33,46,22,35,21,34)(18,24,25,40,27,41,26,32)(19,28,29,43,31,45,30,44)(23,36,38,47,39,48,37,42);(9,11)(10,14)(12,15)(13,16)(17,21,22,33)(18,25,27,26)(19,29,31,30)(20,34,35,46)(23,37,39,38)(24,40,41,32)(28,43,45,44)(36,42,48,47);(11,15)(14,16)(17,21,22,33)(18,25,27,26)(19,30,31,29)(20,34,35,46)(23,38,39,37)(24,40,41,32)(28,44,45,43)(36,47,48,42)

Polycyclic presentation:

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

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, 6467322);