#6484512

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1288 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)(9,10,13,15)(11,16,12,14)(17,18)(19,30)(20,26)(21,27)(22,28)(23,38)(24,29)(25,34)(31,44)(32,45)(33,41)(35,46)(36,42)(37,43)(39,47)(40,48);(11,12)(14,16)(17,19,23,34,24,35,22,33)(18,25,28,30,29,41,38,46)(20,31,36,47,21,32,37,48)(26,39,43,44,27,40,42,45);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,20)(18,26)(19,31)(21,24)(22,37)(23,36)(25,39)(27,29)(28,43)(30,44)(32,35)(33,48)(34,47)(38,42)(40,41)(45,46);(9,12,13,11)(10,14,15,16)(17,21,24,20)(18,27,29,26)(19,32,35,31)(22,36,23,37)(25,40,41,39)(28,42,38,43)(30,45,46,44)(33,47,34,48);(10,15)(14,16)(17,22,24,23)(18,28,29,38)(19,33,35,34)(20,36,21,37)(25,30,41,46)(26,42,27,43)(31,47,32,48)(39,45,40,44)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=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, 6391708);