#6471369

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1660 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

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, 6378565);