#6474051

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1324 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)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,23,31,38)(20,32,35,40)(21,25,33,26)(24,39,41,34)(28,43,46,48)(29,37,30,42)(36,44,47,45);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,28)(21,30)(22,31)(24,36)(25,42)(26,37)(27,38)(29,33)(32,43)(34,45)(35,46)(39,44)(40,48)(41,47);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,20,33,39,22,35,21,34)(18,24,25,40,27,41,26,32)(19,28,29,44,31,46,30,45)(23,36,42,48,38,47,37,43);(1,4,5,3)(2,6,8,7)(11,15)(13,16);(10,14)(13,16)(17,21,22,33)(18,25,27,26)(19,29,31,30)(20,34,35,39)(23,37,38,42)(24,40,41,32)(28,44,46,45)(36,43,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6381247);