#6476619

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470428
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	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1610 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,28,32,43)(20,25,22,36)(21,26,35,41)(24,29,38,31)(30,42,46,47)(33,39,34,40)(37,44,48,45);(9,11)(10,13)(12,15)(14,16)(17,19,22,31,23,32,20,29)(18,24,25,28,27,38,36,43)(21,30,34,45,35,46,33,44)(26,37,39,42,41,48,40,47);(2,5)(6,8)(11,15)(13,16)(17,20,23,22)(18,25,27,36)(19,29,32,31)(21,34,35,33)(24,28,38,43)(26,40,41,39)(30,45,46,44)(37,47,48,42);(11,15)(13,16)(17,21)(18,26)(19,30)(20,33)(22,34)(23,35)(24,37)(25,39)(27,41)(28,42)(29,44)(31,45)(32,46)(36,40)(38,48)(43,47);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(21,35)(26,41)(30,46)(33,34)(37,48)(39,40)(42,47)(44,45)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^2=g,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^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, 6383815);